Honey

By the crude definition of vegan, honey isn't vegan because it's an animal product. However, in spirit this is complicated somewhat by the legitimate and necessary use of bees to pollinate many crops (see Bees in Agriculture), and while most commercial operations over-harvest honey and replace it with sugar or corn syrup, honey can potentially be a legitimate byproduct of strictly pollination-focused operations (see here).

Ethical issues

As veganism is about excluding harmful practices whenever possible, honey is a consideration to be made.
Honey is ethically relevant proportionately to the level of sentience of bees, the amount of bees that suffer as a consequence because of beekeeping, and the environmental impact beekeeping has.

Common practices

Honey-making and beekeeping has quite a few cruel standardized practices.

1. Beekeepers often let bees die over the winter and get new bees for the coming spring--as it's more convenient to simply get new ones, rather than keeping large quantities of beehives safe throughout the winter.
It becomes apparent why that would be the case for big apiaries, where buying new colonies could simply be written off of taxes, and would be significantly more time efficient.
This is not necessarily true for small apiaries, that may keep the bees over years.

These would be the practices required to keep bees alive during winter (rather than simply letting them be wiped out and getting new ones):

• Bees need to be left food stores throughout the winter to survive, or else they'll die of starvation--which is certainly not a pleasant end.
• It's required to keep pest control going throughout the winter for the hives to not be infested by bugs such as mites and beetles, and to make sure to place mouse guards so that rats won't make the beehives their new home.
• Entrances to the beehive are needed to be cleared constantly from snow, since bees need to go out of the hive even during winter.
• Weak hives that are more unlikely to survive would need to be merged with another hive to increase survival chance--a practice that is many times not done correctly, and results in beehives killing each other.
• Some paper needs to be placed in the hive to soak the condensation that forms in the hive during winter (or through some kind of ventilation system), since drips of cold water can make bees die.
• Winds need to be accounted for, placing barriers to prevent the beehive to freeze to death.
• Where temperature fluctuations are strong, beekeepers would need to wrap the hives to keep them warm, or preventively switch all the hives with Styrofoam ones for the coming winter.
• Last but not least, it's important to check on the beehives regularly throughout the winter, to make sure everything is alright, replenishing food stores when needed and managing the moisture level.

All of this would not be business friendly (money committing and time consuming) compared to simply getting new bees.

2. Commercial honeybees are bred specifically to increase their productivity, which narrows down the gene pool of the bee population, and thus makes them weaker against diseases and genetic problems.

This is a serious problem for pollinators, as the diseases born from honeybees can spread to others.

"At present, honeybees suffer from a number of health issues. There are several reasons for that, one of them is that beekeeper practices may not be sustainable. The practices cause a smaller genetic basis for the honeybees, making them potentially less resilient. Use of acaricides, certain breeding methods and migratory practices cause low selection pressure against Varroa infestation and ample transmission of diseases."

"The situation has become even more complex, as beekeepers do not tend to breed themselves, but depend on commercial breeds. The commercial bees are usually inbred strains. The consequence tendency will be that the apiaries are inhabited with genetically more or less the same kind of queens, leading to a lack of genetic diversity within the honeybee populations, limiting the response of the workers to a health challenge or a change of environment or climate. This lack of genetic diversity can lead to diseases having a greater impact on bee colonies." [1]

3. Taking the honey is not healthy for bees and likely contributes to parasites and other problems.
When honey is removed from a hive, it must be replaced with a sugar substitute to give sustenance to bees. A non-honey sweetener is significantly worse for the bees’ health, since honey is the only sweetener that has the essential nutrients that the bees need.

4. There is harm directly done to the bees with the intention of population control.
Wings are clipped off of the queen bee to make sure she doesn't fly away, and killing bees is very common in honey agriculture.

5. Bees are often gassed with smoke machines to harvest honey, as smoking them acts in two ways:

• Smoke masks alarm pheromones that would be released by guard bees with an intruder (or by bees that are injured during a beekeeper's inspection), therefore stopping the bees from gathering the alarm signals from other bees.
• Smoke makes bees believe that there is a fire, and survival instincts kick in. In preparation for leaving the hive to escape the fire, they begin to suck the nectar they have stored and, while doing this, lose their instinct to sting.

Smoke machines are often misused, with beekeepers using unnecessary amounts of smoke and harming the bees. Over-smoking results in bees getting aggressive and/or confused, and having them driven out of the hive.

Environmental impact

While it's commonly believed that beekeeping would be beneficial for the environment (by increasing the total amount of bees), it's often not the case.

Honeybee colonies tend to push away wild pollinators' colonies, and honeybees can't substitute wild pollinators for the environment. Without wild pollinators, wild plants would have a hard time to reproduce (wild plants make up for the vast majority of plants)--wild plants are the plants that grow spontaneously in self-maintaining populations in ecosystems and can exist independently of human action.
Needless to say, wild plants are an essential part to the environment.

On top of this, the amount of bee breeding and the reduction of genetic diversity makes it so diseases are born and are spread to other bees and pollinators (see above), with the potential of causing serious loss of pollination on a global scale.

Health issues

Nutrition

Honey is not a healthy food product. Nutritionally, it is essentially sugar.

Sweeteners like sugar and honey present a variety of health issues, including blood sugar spikes, weight gain (pretty much empty calories), and dental health issues (check the sugar section for a more in-depth explanation).
There are healthier sweeteners, including date paste (whole blended dates), blackstrap molasses, and artificial sweeteners.

However, honey goes one step further with having another health risk.
The difference between honey and sugar is that honey is often contaminated with potentially dangerous bacterial spores which can cause a botulism infection, and is not safe for young children and the immunocompromised.

Medical Uses

If honey had legitimate medical uses, it would arguably be vegan when so used due to medical exceptions.

Commonly, it is used as folk medicine for allergies and cough, and some also believe it has antibiotic properties.

Antibacterial

In-vitro, honey does have significant antimicrobial activity due its hypertonic nature (it dries out and kills cells because it draws water away from them). It may have had some limited use in ancient times where more effective topical treatments were unavailable. In-vivo, orally, it has virtually no effect because dilution with water or stomach fluids negate the hypertonic properties.

Topical use in the modern era is more suspect, but certain kinds of honey are occasionally used for wound healing and may have modest evidence. If FDA approved "Medi Honey" which comes from a specific kind of nectar (from which its medically active component originates; not the bees), is specially sterilized and is prescribed by a doctor for a burn or post-surgical infection, it is of course vegan by way of medical exception.

Allergies

Scientific investigation has consistently shown that honey does not improve allergies.

While in theory very small exposures to an allergen can work as therapy (this is how allergy shots work), there's not even a mechanistic possibility for honey to affect common allergies due to the pollen honey typically contains traces of (remember, honey is not made from pollen) not being the cause of common allergies: even for local honey[2].

Broadly speaking, there are two kinds of pollen that evolved for different methods of dispersal: Insect and wind carried.

The kind of pollen that causes seasonal allergies is the small grained pollen which has evolved to be light enough to be easily carried on the wind, not to entice insects. Honeybees do not typically collect this kind of pollen, focusing instead on larger grained pollen for food.

Even if it were the right kind of pollen, there's no reason to believe it would survive digestion to be able to provide a suitable dose of allergen, or it could provoke an all-out allergic reaction; immunotherapy is based on very careful dosages, and the wrong dose can make things much worse.

If you need help with allergies, use conventional antihistamines & steroids as prescribed by your doctor, or allergy shots.

Cough Suppression

Cough suppression is complicated by the fact that there are very few substantially effective medications available, placebo control being difficult for many remedies, and cough suppression itself being highly subjective. Several options outperform placebo, but only by a small margin.

On cough suppression of Honey, a Cochrane review concluded:

Honey may be better than 'no treatment', diphenhydramine and placebo for the symptomatic relief of cough, but it is not better than dextromethorphan. None of the included studies assessed the effect of honey on 'cough duration' because intervention and follow-up were for one night only. There is no strong evidence for or against the use of honey.[3]

In other words, it's possible but there's no good evidence. It's important to note that until very recently there was no good evidence that diphenhydramine (and antihystamine) was effective for cough at all (it does have a small effect at suppressing cough from capsaicin[4]), and even dextromethorphan (which is better than honey) is probably only slightly better than placebo.

If you have a cold, there's no credible reason to turn to honey to relieve your symptoms. There are barely superior medical options, but that's only because those options are terrible[5], not because honey is good.

If you feel like they help you, your best bet is probably non-medicated cough drops with sucrose and menthol, which decrease sensitivity to coughing stimuli[6](the numbing of menthol helps likely reduce sensitivity, the effect of sweetness is less clear).

Bees in Agriculture

Some important questions to ask may be if these plants are really necessary to eat, and how they compare to other crops. There are a lot of unknowns in insect mortality in agriculture, as well as other questions of efficiency and sustainability (some are discussed in the article on Sustainable Vegan Agriculture), and while these unknowns probably do not justify any kind of massive shift in agricultural practice, the known harms may justify easy changes in consumer ones when it comes to honey.

Bee Sentience

Bees are one of the most sentient insects, with each bee having an average of 1 million neurons (if you're not sure how that's relevant to the degree of sentience, you can check the sentience page).

Bees have been shown to learn through operant conditioning (learning behavior through reward systems), which is the smoking gun for proving sentience.
One of the reasons why they have evolved with higher cognitive capabilities than other insects might reside in the fact that they're very motile, and they have to quickly understand and process the stimuli around them to know if something is a threat or not and to know where to go, recognizing flowers and many other environmental factors that might be of interest to them to avoid or to check out for pollen - on top of having to recognize other members of the nest and being able to communicate with them.
The fact that bees are social animals is also a reason why a higher degree of sentience than other insects would make sense - they can recognize features among themselves, and even recognize human faces. [7]