Will helium in the tires of a bike make it lighter?

It will make it lighter, but the effect will be very small. The volume of the tube is probably less than a liter. One mol of an ideal gas is 23 liters at atmospheric pressure. So you have about 0.2 mol of gas in there at 4 bar pressure. Helium weighs 4 g/mol, nitrogen about 28 g/mol. So for 0.2 mol, the weights are 0.8 g and 5.6 g. Cleaning off the dirt from the frame will have a greater effect.

Helium atoms are smaller than nitrogen molecules. Therefore there is a greater rate of diffusion through the bike tires. Your tires will become flat quicker than normal. Therefore it is not really a good idea to use helium.

Helium has been used in racing bicycle tires for indoor track (velodrome) events.

The helium will decrease the overall weight of bike and rider only slightly, and it will somewhat reduce the angular momentum of the tires.

There's also the possibility that helium is more "elastic" than normal air or pure nitrogen, which would reduce rolling resistance, but I haven't found any articles about this. Helium is slightly less compressible than air, but I don't know if this affects "elasticity".

Depending on tire technology, the helium leaks far too quickly to be useful for a longer period event such as a road course. This could change if tubeless tires could be made to not leak helium. I don't know if the rules for bicycle racing have changed to disallow helium in tires anymore.

Update - there may be another factor, heat dissipation.

"We used helium because it is such a good heat conductor. Our rims and tires were so thin heat buildup was a hazard and the helium helped keep them cool."

Scroll down a bit into this forum thread:

I recall decades ago someone mentioning that using helium allowed somewhat higher pressure to be used in the track bicycle tires, which could be related to the heat dissipation, but I haven't found an article to confirm this yet.

The most common wheel size for mountain bikes is a torus with R = 307mm and r = 27 mm. Assuming about 7mm of rubber on the rolling surface, this amounts to about 2.4 liters of volume. For a typical pressure of 70 psi, you'll pump 11.5 liters of air in your tire. That air would have a mass of about 14 grams, and an equal volume of helium would be about 2 grams.

For both wheels, replacing air with helium would give you about 24 grams of weight reduction. Not really worth it, unless you're about to set an Olympic record and every gram counts.

Also, if your bike had tires big enough that filling them with helium would make the bike float, it would float upside down ;)