Does water boil faster with the lid on?
Yes, water does boiler measurably faster with the lid on.  

The reason is simple:  in order to boil, water must be heated to the boiling point (okay, that was obvious).  However, while heat is being introduced at the bottom of the pot, heat is also being lost at the top of the pot, through three means:  evaporative cooling, and air convection of heat away from the surface of the water, and radiation from the surface (this last is probably the least significant).

Its like going outside with your hat off--you need to work harder to stay warm, because you are loosing heat faster.  Similarly, the flame has to introduce energy to make up for the cooling at the top of the pot.

With the lid on the pot, convective cooling will be substantially curtailed, as cool air will not be able to circulate directly onto the surface of the water to cool it; radiative cooling will be slightly reduced, because heat will have to radiate or convect to the lid, to then radiate from there.  

Additionally, with the lid on the pot, the evaporated water cannot diffuse or convect into the room.  It will soon reach vapor pressure equilibrium and begin condensing almost as fast as it evaporates, returning much of the latent heat of evaporation as almost as fast as it is lost (it is not a total recovery, because the pot with lid is not air tight).  

So the largest of the cooling effects will be curtailed.

The hat is on the pot, as it were :-)

This article at Mind Your Decisions shows actual home experimental measurements.  Boiling 16 ounces of water:


  The covered pot boiled quicker at 4 minutes and 15 seconds. The
  uncovered pot took an extra 30 seconds to boil at 4 minutes and 45
  seconds.


You will have to decide if the time difference matters to you.  It certainly will cut your electric or gas bill a small amount.

It is the reduced cooling--essentially--insulation like a hat, that is the true effect of the lid.