Mike as fas as burgers go, the only reason not to eat them right away is to let them cool down so you dont burn yourself.
It matters on the temp too, if you get a med rare and its all juicy, all its going to do is soak into the bread and make the bread soggy, which is why even at home i always grill/toast the bun, and one of the main reasons restaurants do it.
As far as steaks/roast/turkeys go the theory behind letting them sit is when there cooking all those juices are piping hot and flowing thru the meat, letting it sit always the juices to cool and settle back into the meat, as they carry a lot of flavor and youll end up with a juicer cut.
If youve ever cut a steak right off the grill, youll notice most the juices run out, but if its just you then who cares.
If youre cooking a steak past Med rare in my opinion its a waste anyway, i hate when people come into work and order kobe burgers well done, or tuna steaks well done, just eat canned tuna then, the worst was when I ran an italian restaurant and people would order our 24oz cowboys steak well done then complain theyve been waiting 45 minutes.
here is a piece from a meat cooking article, simple terms.
"I've always been told that this happens because as one surface of the meat hits the hot pan (or grill), the juices in that surface are forced away towards the center, increasing the concentration of moisture in the middle of the steak. Once the steak gets flipped over, the same thing happens on the other side. The center of the steak becomes supersaturated with liquid—there's more liquid in there than it can hold on to—so when you slice it open, all that extra liquid pours out. By resting the steaks, you allow all that liquid that was forced out of the edges and into the center time to migrate back out to the edges."
makes sense.