Weight gain.
Not really a complication per se, but an often complained about sequelae of desexing surgery.
Studies have shown that neutered animals probably require around 25% fewer calories to maintain a healthy bodyweight than entire male animals of the same weight do. This is because a neutered animal has a lower metabolic rate than an entire animal. Because of this, what tends to happen is that most owners, unaware of this fact, continue to feed their neutered male dogs the same amount of food after the surgery that they did prior to the surgery, with the result that their dogs become fat. Consequently, the myth of automatic obesity has become perpetuated through dog-owning circles and, as a result, many owners simply will not consider desexing their dogs because of the fear of them gaining weight.
Author's note: The fact of the matter is that most dogs will not become obese simply because they have been desexed. They will only become obese if the post-neutering drop in their metabolic rate is not taken into account and they are fed the same amount of food calories as an entire animal. Any weight gain that is experienced can be reversed through not feeding the pet as many calories and treats.
Preputial scalding and infection - a potential complication of early age desexing.
Normal testosterone levels are required in order for young male dogs to be able to break down the adhesions between their penises and prepuces (penis sheaths) and therefore extrude their penises. Animals that are desexed very young (i.e. early age desexing) may not be capable of extending their penises from their penis-sheaths at all. These animals are therefore prone to having urine pool within their prepuces during urination, resulting in urine scalding of the prepuce and secondary preputial infections and discharges. Vigilant hygiene and cleaning of the prepuce is needed in these animals.
Neutering didn't deliver the change (improvement) in problematic male behavior that you thought it would (e.g. behavioural problems such as aggression, dominance, marking territory and roaming have persisted despite desexing surgery).
Again, this is not really a complication per se, more a problem of unfulfilled expectations. A lot of owners only get their animals desexed as a means of trying to correct already established male behaviours that are annoying or unsafe to the owner or pet (e.g. roaming, humping legs and toys, aggression, dominance, marking territory and so on). Such owners often become very disappointed when the desexing surgery fails to correct these behavioural "defects" in their animal.
The trouble with having such expectations is that, while desexing a dog early may often go a good way towards preventing these adverse male behaviours from developing (one of the reasons we advocate early neutering), once these behaviours have become well established in the dog's behavioural repertoire (i.e. become part of its character), they are often difficult to reverse by desexing alone. The reason for this is that, over time, these adverse behaviours go from being purely hormonally-driven behaviours (i.e. treatable by removing the hormone responsible: testosterone) to learned behaviours (which are hard to unlearn). For example: a dog might initially start roaming the countryside because of a hormonally-driven imperative to find females, however, over time, such a dog will soon learn that roaming the countryside is fun and will therefore continue to do so, regardless of whether it has testicles or not. Dominance behaviours are also learned: a young male cat or dog might start pushing the leadership boundaries in your household because of testosterone, however, it will also learn what it get away with by the way you, the owner, react to its pushes for leadership.
Another major reason why desexing surgery often fails to deliver improvements in a pet's behaviour is that the behaviour being 'corrected' by surgery is not a testosterone-induced behaviour! Many "bad" behaviours have nothing to do with testosterone and are a result of poor socialization and bad training and lack of "owner leadership" - thus, removing the testicles won't cure these problems. Aggression is a good example of this. Many owners with aggressive dogs try to cure the aggressive behaviour by desexing their animals. The trouble with this is that the aggressive behaviour may or may not be being caused or contributed to by testosterone and, thus, desexing may not help matters. For example: fear-induced aggression (fear-biting) is caused by past experiences; owner reinforcement (e.g. owners that pet and soothe snappy, fearful dogs in the veterinary clinic are often making the aggression problem worse because the dog is getting rewarded for the nasty behaviour by being petted) and the emotional nature of the dog (this has a bit to do with the dog's genetics, breeding and the temperament of the dog's mother and father), not by testosterone. Desexing will not fix fear-biting. Inter-male aggression (dog-to-dog aggression), however, is contributed to by high testosterone levels and desexing dog-aggressive entire males may go some way towards helping the issue, provided it hasn't been going on too long and the animal hasn't now 'learned' to dislike other dogs.
Author's note: certain testosterone-mediated behaviours like inter-male aggression and hypersexualised behaviours like mounting chair-legs and toys may not resolve with desexing alone because of presumed alternate-sources of testosterone production in the body. Although the testicles are the major source of testosterone in the body, other glands (e.g. the adrenal glands) and even parts of the brain may produce it too, resulting in some testosterone-mediated behaviours persisting despite desexing. The mounting of chair legs and toys is an excellent example of this - it is a hypersexualised behaviour and although many cases are 'cured' by desexing, the problem often persists despite desexing surgery.
Animals whose testosterone-mediated behavioural problems have not resolved after desexing surgery may be able to be treated with the administration of "anti-testosterone" drugs such as MPA (medroxyprogesterone acetate), megestrol acetate (tradenames include Ovarid) and delmadinone (tradenames include Tardak). These progesterone-based drugs oppose the effects of testosterone in the body and effectively neutralise the effects of testosterone coming from alternate sites in the body. They can produce excellent results in the management of hypersexualised behaviours such as chair-leg mounting and toy humping.