When are pot lids useful?
Covering a pan slows down evaporation of the water. As evaporating water takes a lot of heat, covering a pan will make the contents boil earlier. 

If the recipe tells you to cover the pan, it's to make sure there's enough moisture for the full duration of the cooking; without the lid, you might end up with a dry (and usually burned) mess.

When cooking rice, the proteins and starch dissolving in the water will cause the bubbles that form on boiling to be more resistant, so they form a foam that lifts up the lid and escapes. Usually after the first foam is destroyed, you'll have no more problems (but do keep the fire low). And for at least one method of cooking rice, you want to keep the pot well covered, as the amount of water you add is just enough to fully cook the rice (the water is absorbed by the rice and makes the starch grains swell, without water the rice would stay very hard).

It all boils down to how fast you want the water to evaporate.