If I pumped up my bicycle tires with helium instead of plain air, what would happen if the applied force on my pedals was constant?
Would I go faster because of the reduced ground friction? Would I go slower because I would have less contact with the surface I'm riding on? Or would it make no difference?
It wouldn't make very much difference. Let's quantify how much.
First, estimate the volume of a bicycle tire. Most you can encircle with your thumb and forefinger, which gives a cross-sectional area of order $\pi\rm\,cm^2$. A "700c" wheel is named for its diameter, about $700\rm\,mm$, so the circumference is a couple of meters. That gives a bike tire volume of about $600\rm\,cm^3$. There seem to exist mountain bike tires with diameter five inches, so the high end of tire volumes is thirty or forty times this large, perhaps $24\,000\rm\,\rm cm^3 = 24\,liter$. Let's use this volume.
The buoyant force exerted by a fluid is equal to the weight of the displaced fluid. The mass of $24\rm\,liter$ of air is about $24\rm\,g$. This isn't very much lift compared the the mass of a typical cyclist. In fact, switching from typical rims/tires (which was saw would provide roughly $0.6\rm\,g$ of lift, if you'll forgive me providing a force with mass units) would probably increase the mass of the bicycle by more than $24\rm\,g$, so you'd come out heavier rather than lighter.
A bicycle tire should roll along the ground without slipping. (Rolling friction comes from the wheel's motion around the hub, and to a lesser extent from deformation of the tire near the ground.) If you could provide a significant amount of lift, the tires would be more likely to skid during hard starts and hard stops, but rolling friction wouldn't be very different. Providing this much lift in the tires would make the bike vertically unstable --- it'd want to flip over, with the tires in the air and the cyclist on the ground. In such an inverted configuration, you'd go slower and find the ride less comfortable.
Others mentioned that the weight difference will be small, on the order of 10g (one should take into account the pressure in the tires). You should have in mind though that lighter tires also have smaller moment of inertia (rotational inertia), which should facilitate acceleration. Some people also mentioned helium's higher thermal conductivity, which supposedly might help prevent heat buildup (http://www.bikeforums.net/road-cycling/331014-helium-your-tires-truth-myth.html)
