Do burgers cook better when close or far apart?
On the grill, most of the heat will be coming from underneath the burgers (you already knew this)

Having a gap between them will let air flow between them and increase the temperature around the meat.  If they are touching, the airflow will be hindered and you won't get even cooking.  When the burgers are all squashed together, you're effectively cooking one big burger and not many individual ones.

ie)  The spot where the meat touches will cook slower than the outsides where there is now more heat.

Purist grillers will tell you that the "best" way to cook on the grill is with the top open, if you do this, you'll get even less heat in bunched up areas.

Separating them more than an inch apart won't really affect cooking time.