Will adding white vinegar to toilet tank dissolve lime buildup?

Do you mean your toilet tank leaks and then fills up during the night? Probably need new "inside works" to keep it from happening. If you think it is just lime, then ask your hardware store and they will have something for that. Vinegar will disolve lime on faucets but not sure about a toilet tank.

Sounds like the flapper is not sealing properly. Make sure there is some slack in the chain so the flapper seals. Replacing the flapper would help. There is barely any chance of lime build up in the tank since water flows with every flush and doesn't sit long enough to for any.

http://www.housecleaningcentral.com/en/cleaning-tips/bathrooms/cleaning-inside-of-toilet-tank.html

Sure can! https://www.youtube.com/watch?v=2cL0Jq6dQIA