Should door of oven be kept open or closed while grilling?

A grill is designed to heat the food from one direction; I'd imagine from the top in your case. If you close the door the heat builds up inside and it just cooks the food like an oven would.

Since it is not designed to do that (ie. there's no thermostatic control), there's also a chance it could overheat and become damaged.

Some more modern grills do not need to be left open whilst in use. These use a fan to extract the hot air building up inside, and draw in cool air.

Refer to your instruction manual if in doubt.