I took off the tank protector that came with the bike, and noticed a small chip in the paint under it.  Put on a new protector not thinking much of it...will this chip rust significantly, or will the tank protector more or less seal it?

The tank protector will collect water. So yes it will rust. However you may not have to worry ... cos the crap pockets will rust out first anyway.
Cool.
Buddha.

Probably should have hit it with a primer of some sort, but the glue from the protector should help...kinda depends on the seal between the tank and the protector...but some moisture will probably get in there...

Later.

I guess when I have the chance I'll order another tank protector and seal it.

I had a clear tank protector once, and you could see a lot of dirt/gunk collecting right along the edged, but I didn't really see any get under the main part of it. I rode in all kinds of rain too, so assuming the chip isn't right at the edge, I think you'll be ok for a while. Just get it with a little bit of nail polish before you put the next one on.

My tank had a 1/8 inch chip down to bare metal when I bought it two years ago. The chip is on the side of the tank and never covered. There has been no evidence of rust since I bought it.

I took the existing tank protector off, cleaned the chip, and covered with the white touch-up paint that I have.  New tank protector going on when it arrives.