I almost always cook on my Webber Charcoal grill with the lid closed. One word of caution, if using charcoal; If the meat is very fatty, the closed lid (with vents open) can concentrate too much smoke, which is deposited as particulate matter on whatever you are grilling. Though that smoke is what gives grilled meat that wonderful flavor, if there is too much of it, the excessive smoke can create a bitter, almost inedible flavor. This happened to me once, when I added extra beef fat to the grill as I thought the meat had too little fat to create the smoke. The particulate coating could litterally be rubbed off with your finger, and had to be washed off of the meat, which removed much of the seasoning as well. The meal was pretty much ruined.
Normally, the smoke generated by the meats own fat, dripping onto the hot charcoal is just the right amount.
If you are using wood smoke to help flavor the grilled food, then covering the grill is a must.
Large chunks of meat need to have the grill covered, and sometimes with the vents half-closed to control the heat by starving the fire of oxygen. I do this with crown roasts, brisket, and with whole turkeys, game hens, chickens, etc. The grill then doubles as an oven, creating the hot environment to cook the meat through, and creates flavorfull smoke.
Seeeeeeya; Chief Longwind of the North