I had no idea. I honestly had no idea how much weight it would save, so I did some math. I'll post all of my work in a pdf file you can download at the end of the article. But what I found out is that, assuming you have a 2" tire (Kinda just picked a middle ground and easy number to deal with*), and ride a BMX bike*, and ride it at 90PSI (an average, I know people go higher and lower), the air in your tube weighs in at a whopping 0.03458 pounds.
But what would it be with Helium in there instead of Air? The helium in the tube would weigh in at a total of .004778 pounds.
Now, is that a large difference? Well, the weight of the helium is actually only 14% of the weight of the air. Granted the weight of the air is small, but it is still relatively significantly lower.
These numbers are not perfect. They assume a lot of things and factors to be true or false, and in real life, these numbers probably wouldn't be true (though they are a nice approximation).
There you have it. Do with the numbers what you want. This was not done to prove a point, only to investigate.
Also I would like to note: Helium is small. And by small I mean the molecules are smaller than the "air" molecules that would be in the tube. The tube is porous. Everything is porous, though some materials may be more/less porous than others. The tubes will leak through both the valves and the walls themselves. The helium will drain faster.
And another note: These weights are dynamic. They are moving. Your wheels are both spinning about their axles and are moving about when you're in the air. The effective weight of the air in the tubes is different than the static weight.
*If you look at the work, at the bottom of the page is actually a formula that I derived for use with any tire width and diameter.