This is one of those situations where you could argue on and on about definitions so let's answer it in all three meanings of the word weight that people might have.
Does your mass $m$ increase when you inhale helium?
When you fill your lungs with helium, you transfer $\approx1g$ of matter, which is several times less, but roughly of the same order of magnitude as when you fill your lungs with air. Those amounts are relatively negligible for all practical purposes, but strictly speaking, when you fill your lungs with helium you do gain mass compared with you having empty lungs (obviously).
Does your weight $F_g=mg$ increase when you inhale helium?
Obviously, for the same reasons as above, but still negligible.
Do you "feel" lighter, or more precisely, does your weight scale read a smaller number when you inhale helium?
This is where we take into account that we're not in vacuum, we're surrounded by air and we have to account for buoyancy. But since the buoyancy force is equal in magnitude to the weight of the displaced fluid, it doesn't matter whether we inhaled air or helium, because in both cases we've displaced air.
The number on the (extremely precise) weight scale will go down (in comparison with you having inhaled air!!), but for the exact same reason as in the previous two situations and it will also be negligible!
This might sound wrong because "helium floats", but think about it... When you're hugging a huge helium balloon, you're increasing your total volume (you + balloon) so you're displacing a significant amount of air without adding any significant mass. In that case the number on your weight scale should drop. (It would have to be a pretty huge balloon, but you get the point).
If this still doesn't sound right, consider a situation in which all of your internal organs, tissue, bones and everything except a thin layer of a skin is replaced by helium with the same total mass (disturbing, I know, sorry about that!). Assuming your body temperature was somehow the same and not being worried about the fact that the pressure inside your body would be several thousand times higher than atmospheric pressure, your weight would be exactly the same, and so would your scale reading. Why? Because the thing that matters when considering buoyancy is the density of the fluid displaced and that's always simply the air around you.
I hope this helps, any suggestions for improving the answer or comments are welcome :)