If you get a hardship withdrawal to pay a dependent's college tuition, will you have to prove that you used the money for that tuition when you file taxes? What if you don't use the money for tuition?

You have to prove a hardship to get the withdrawal. Having done that, you do not have to spend the money for that hardship. The distribution is taxable regardless of what you do with the money.

The easiest proof of that position is that the portion of any hardship that is not pretax ee contributions is rollover eligible. So you can get a hardship distribution to prevent foreclosure on a primary residence, and roll the distribution to your IRA.