False.
Sort of.
This all depends on your point of view. If you're talking about pure, contaminant-free (this includes air bubbles as well) H2O, then no, hot water does not freeze any faster than cold water. In fact, in this case, it freezes quite a bit slower.
On the other hand, if what we're discussing is straight-from-the-faucet tap water, then it will freeze faster (and stronger, I might add) if it's heated prior to being placed in the freezer. Then again, tap water will also freeze faster if you heat it, let it sit for a while, and then freeze it.
But why, you ask, do these inconsistencies exist? As mentioned in Natrous's writeup, heating water allows gasses to rise to the surface of the water and escape. In fact, these kinds of gasses (and other impurities) are prevalent in all water to some degree or another excluding the Platonic ideal discussed above. These tiny bits of gas act as insulation within the water, much as pockets of air do in foam. Heating the water causes the H2O molecules to move about quite a bit, shaking up these gasses (present even in distilled water, which can trap air bubbles due to being shaken about) which then rise to the surface and escape. The key here is that it isn't the heat that causes the faster freezing time, it's the release of gasses. The water doesn't need to be hot to freeze faster, it needs to have been heated at some point prior to being frozen. Which, of course, is more or less the same thing, but the distinction, however slight, needs to be made.
Boiling the water causes the most satisfactory ice, since the water is at it's most severe state of agitation. In fact, if the water is boiled without heat (such as being placed in a vacuum), the resultant ice will freeze just as fast and strong as if it were boiled using a stove.
This whole mess is due to the human tendency to simplify things, such as Occam's Razor. When people see something heated freezing faster, they naturally assume that it's the heat causing the accelerated freezing. When it was later discovered that boiled water froze faster (this is the version I heard as a child), it was again taken for granted that it was the heat increase responsible, not the boiling action itself. Since heating the water is correlated to a faster freezing time, it can be a tough time convincing someone that heat is not the cause.