People have seriously considered it and tried it. I blame the people at my Hyundai-related discussion forum for costing me an hour reading about helium.
Yes, you can put helium in your car or bicycle tires. Yes, it would be lighter than air or nitrogen probably by a few dozen grams per tire. You would save (much) more weight by tossing out the contents of your ashtray/change drawer, leaving your sunglasses at home, washing the dirt off your car or having a piss. Also, it is the smallest element and would leak out at annoyingly high rates from your tires.
Helium is also a limited-availability natural resource with a huge price distortion caused by the USA government. Currently it is cheap because we are selling our stockpiles**. Market forces will react when our sale ends, and the price will rise probably dramatically, but probably we will still be able to afford helium balloons for parties. I stopped buying helium balloons for parties when I figured this out – I’d rather have it cheaper for longer, to keep down the cost of MRI testing where people currently NEED helium (as a coolant) to save lives.
It used to be hard to make and we didn’t have gobs of it laying around. It was $2500/ft3 in 1915 and $0.15/ft3 in 1989. One isotope is currently still at $2000 because they make it from Tritium. Helium is also mined with natural gas, and the prospect of “running out” of helium is something our great-great-etc. grandchildren will have to worry over, if anyone will. 100-300 years-ish. As it gets more expensive, look for more efficient uses in imaging as well as higher incidences of recycling (vs. blowing it out to the atmosphere). Also, expect lower-cost practical applications of MAGNETIC cooling. (mind = blown)
The 1984 US olympic team allegedly used it to conduct heat out of their tires (helium conducts heat better*). You can save 10 or 15 grams per tire using He but when you already have to ADD ballast to make minimum weight on a real race bike, this is not the best use of your time/money.
the Skunk Works allegedly tried and declared it to be too much hassle, even with military-ish maintenance schedules, a bottomless budget and a desperate desire to lose weight on the SR-71.
At least *some people* at Lockheed have considered it and figured they could save 3 lbs. per aircraft. And maybe the tires could explode (but that was speculation).
**stockpiled to float our national security related blimps early in the last century, and not strictly required for exactly that these days.
*and supercooled helium doesn’t/can’t boil. The way it transfers heat means it goes straight to gas instead of boiling internally. Crazy.