FRE0 saidA. It will definitely boil faster with the lid on, but assuming that the burner puts out plenty of heat, the difference will be small. Unless you actually accurately time it, you will not notice the difference. Also, determining the exact instant at which boiling begins will not be easy so probably you could not accurately make the comparison anyway. If you try it, be sure to measure accurately the amount of water you use and be sure that the starting temperature is the same.
To be really scientific, you will need to repeat the experiment several times and tabulate the results.
Since water boils at different temperatures depending on your altitude I was thinking that you could bring a pot of water to boil and measure its temperature with a digital thermometer and use only the temperature as the point where you consider it boiling.
In a laboratory setting you could have a digital timer connected to the on/off switch to your heat source so that the timer starts when the heat is applied, and the digital thermometer is connected to the timer, or everything is connected to a computer which is logging the collected data (every millisecond, for example). In that setting you would certainly see that with the lid it boils sooner. Maybe only a few hundred milliseconds sooner, maybe even a few thousand.
In a kitchen setting it's very likely irrelevant since you're probably busy cutting and chopping or doing something and won't notice when the water starts boiling. And if you always use a lid I doubt if you'll see any reduction in your gas/electric bill.
It was a slightly rhetorical question, originally asked by a roommate's girlfriend many years ago. He and his girlfriend were physics majors. At the time I remember thinking it was a silly question and that the lid wouldn't make any difference but I later realized that it would make some difference but it'd be too small to matter in practice.
I started thinking about it again when I got an induction stove; with the induction stove it's really irrelevant since the water comes to boil so quickly that I'm still amazed.