The correct answer, of course, is it depends.
Gas grills cook mainly by convection (i.e. hot air) with just a little radiation mixed in. The lack of radiation is why high end gas grills will have that separate IR burner as a feature. Keeping the lid down on your gasser lessens the huge amount of convection energy loss that you get with an open lid. You'd never cook with a kitchen oven with the door open, right?
An uncovered charcoal grill (like a hibachi) cooks almost entirely by radiation -- light energy like you get from the sun. To get the most radiation energy into the food, you'd want your food to be very close to the coals (like with a hibachi). Radiation is intense -- it is great for searing/browning. But also easy to overdo. Putting a lid (as with a Weber kettle) on a charcoal grill retains heat and adds a convection capability to a charcoal grill. If you want your charcoal grill to cook via convection, use an indirect zone and put a lid on it.
Now that we have the science down, how should you cook your burgers?
If using a gasser, always keep the lid down. Otherwise you are just wasting time, fuel and convection energy. To get more sear/browning, the lid makes little difference. But you'll need to do something extra. IR burner, maybe some flame kiss from dripping grease, or some hot metal (grates, griddle, cast iron pan). Since hot metal touching the food (cooking via conduction) is the most efficient browner of them all. On my gasser, flat side grillgrates are my searing tool of choice.
On a charcoal grill, putting the food close to the coals is what gets you the best browning. Lid does not matter, unless you want to cook faster by getting the convection going too.
But if your burgers or steaks are thick, you will not want to cook only by direct radiation. Which leads to incineration. Which is why you'll go convection at some point -- indirect zone, a lid, or moving the food further away from the coals (i.e. the height adjuster on a hibachi or a Santa Maria grill).
This is how all of us cook on our grills (direct/indirect; if you are looking you aint cooking, etc.) even if the science isn't consciously guiding our methods.