The lid can do one of three things: Keeping the lid down increases the temperature of the air in the grill.
This means that in addition to cooking with the direct heat of the coals, you are also cooking with convection of the air.
This won't happen with the lid up.
In a charcoal grill, the temperature the coals burn is based on the airflow through the grill.
Open is as hot as it will go (assuming the bottom vent is also open).
By closing the lid and altering the airflow, you can control the temperature of the grill.
My grill can keep fine temperature control between 200F-700F depending on how you set the airflow.
Which leads us to smoke.
It seems obvious that if you are smoking, you will impart more flavor by trapping the smoke with your food rather than letting it float in to the sky.
Additionally, you want wood to smolder not burn for your best smoke production.
This works better at a lower temperature (see #2).
cooking.stackexchange.com/users/1259/yossarian
Keeps the Dog from stealing the meat on the grill!
cooking.stackexchange.com/users/105/BillN
It keeps the heat in, so you can switch techniques from a pure grilling technique (Open Lid, hot coals = direct radiant heat) to a baking/smoking technique.
For example, when I do a duck breast, I put all the heat on one side of the grill (i.e. 2/4 gas burners, or pile the charcoal on one side).
I sear the breast skin up directly over the heat, and then flip it to brown and crisp the skin.
Once the fat starts to render, I get flare ups that are completely un-manageable.
The solution is to move the breast to the cool side, and shut the lid, so that it bakes and smokes for a few minutes to finish off the cooking.
cooking.stackexchange.com/users/1148/Chris Cudmore
Like the people say, open lid means directional heat from the coals.
You'll cook the bottom of your food much faster than the top.
You'll also get more air to the coals, so they'll be hotter, and to the food, which dries it out faster (at least in dry places like Calgary, where I live).
Closing the lid, keeps the heat more 'omnidirectional', cooking from all sides more evenly.
And generally keeps the moisture in.
All evidence of this is from my cooking (on stoves, outdoor grills, campfires) and from what I've picked up from my family and friends.
Take it as you will.
cooking.stackexchange.com/users/501/Scott Newson
I am assuming you are talking about a charcoal grill.
When grilling over direct heat, using the lid serves two functions:
First, your food will cook, albeit to a far lesser degree, on the sides that are not facing the coals.
Putting the lid on will trap heat in your grill, providing convection as well as direct cooking.
Also, you will be providing less oxygen to your coals with the lid on.
This is key in controlling flareups.
This is the more important benefit.
The biggest risk to ruining your grilled food is the intense heat from flareups on your grill.
cooking.stackexchange.com/users/2832/Sean Hart
I have just bought a new half drum barbeque which can with a fold down lid.    I have heard that these lids are good for creating a smokey flavour, or for keeping food moist whilst it cooks. A lot of this seems to be based on peoples personal opinion with evidence based answers about what the benefits of cooking with the lid down are.   Does anyone have any evidence of exactly what cooking with a lid down on a barbeque actually does?
What are the advantages of a lid on a barbeque?
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cooking.stackexchange.com/questions/4329
cooking.stackexchange.com/users/126/Ian Turner
The lid retains the heat and increases the air temperature in the grill, allowing food to be baked/smoked rather than just grilled. Keeping the lid on increases and retains the temperature in the grill, which is useful for smoking.
Keeping the lid on increases and retains the temperature in the grill, which is useful for smoking.