Title says it all and this is not just a burger question, what about those steaks, chicken breasts, pork chops on the grill?
I believe the goal is to get the best sear possible on each side of the meat. Moving the meat around too much could cause it to cook completely without getting a good sear on the outside.
Clearly this doesn't apply when doing anything "rotisserie" style, as you're trying to do the exact opposite: slowly cook the interior of the meat without burning the outside to a crisp or unevenly.
Also, see this similar question: Grilling burgers: flip once, or keep flipping?
I think this rule really applies to the grilling marks on the food. The longer you leave the food on the grill the more those marks burn themselves into the food. If you want the pretty "clean" lines/pattern then don't move it.
I actually prefer, in the case of meat, to get an even sear on the meat's surface, so I actually move it often to not get the pretty pattern. So the whole thing looks more charred than having the pretty lines on the meat.
In some cases I actually finely score the surface of the meat to be grilled so I get deeper char marks into the meat. But that's just me. Also you will get more marks if the item is dry vs. wet or moist.