Interesting, well seeing as I have been asked for an opinion, here goes.
I personally have only ever taped once, and that was after I landed on someone's foot. I believe in strengthening the body rather than strapping parts to hold it together. I regularly exercise weaknesses, and for that period of time I spent the bulk of my training on the peroneals etc to help regain some subtalar stability. I have never had problems or the need to brace/strap since. I consider myself somewhat lucky, but also consider partly due to a solid rehab program rather than just taping and playing, probably only spending a month strapping total.
The argument that tape/bracing weakens the ankle is often used, and I somewhat subscribe to this argument. This misleads some people, as simply putting tape on does not instantly weaken the ankle. The body has the capacity to adapt to stresses placed on it, and does not like to be wasteful. Muscles use energy, and if they are chronically not being used, they will break down (atrophy). Think about any time you have been bed ridden for an extended period of time, you lose weight (muscle mass), as it is not needed to just lie there, so the body adjusts. It is a similar concept for site-specific muscles, if you do not exercise them, they will atrophy. Thus in the ankle if you strap all the time, you get some support from the tape, so the body does not need as much stability from the muscles themselves so they adjust, and the athlete becomes reliant on tape. It is for this reason I will choose never to tape myself (unless injured), but rather strengthen and condition the appropriate muscles to maintain a solid ankle. "But what if you land on a foot?" yes that is the risk I take, but with better ankle proprioception through training I would suggest you would be better off anyway.
142232 it is not the total time your ankle is taped that is important, I think your 22 hours a day argument is not valid, as when you are sleeping your ankle is under no load anyway (unless you sleep standing up). However the load you place your ankle under, if you tape during all exercise, then the total load is less. Consider going to the gym and only ever bench-pressing the bar, as opposed to bench-pressing more, the more load the greater the gain (obviously under injury threshold load in the case of the ankle). I liken walking around all day untapped, in this example, to the bar with no weight, and training untapped like the bar with weight, I hope that makes sense to you.
142221 is right though after a serious ankle injury it may never be the same. I would suggest there are many variables as to why you may tape. But as said my preference is against strapping or bracing, and one for conditioning. If there is someone who has chronic ankle instability, or has a lax ankle, there may be a case for strapping/bracing, however for the majority of people it is unnecessary in my view.
A few things to consider may be that when strapping, who is strapping your ankle? If you/they are not confident that they can strap, then maybe look at a brace. However braces come in a few sizes and may not necessarily be correct for the individual. Also I believe most of the time ankle injury occurs to the anterior talofibular ligament when the foot is in plantar flexion and inversion, do braces limit this, or do they only limit inversion/eversion? As well as other valid points previously put forward by others above; I do not feel the need to repeat.
In summary I feel subtalar stability can be maintained without the need to brace or tape, but if there is a special case where it is needed, then either method can be used if joint mortise is able to be maintained, there may be different reasons why you choose either, considering the injury mechanism, and what structures are involved will play an important part in your decision. I choose neither.