Do out of state college students have to file income taxes for their school state and home state?

Yes, if they earn money in another state they will need to file a return for the income earned in that state even if they are a student. 

Also, since their home state is still there home as school is considered a temporary absents, they will need to file a return in their home state.  So they would file a nonresident return in the state they are attending college and then a resident return claiming the same income in their resident state but then they will claim a credit on their resident state return for taxes paid to the nonresident state(school state)

This is true for most states, some states may have slightly different residency rules so if you reply with the states, we can give you the exact answer or you can also just walk through both states returns and answer the questions and TurboTax will determine if it is a state that is an exception to the rule. 

Q. Do out of state college students have to file income taxes for their school state and home state?

A. Yes, both.

If you have a reciprocal state situation, the answer is different. You would only file in your home state, unless you needed to get school state withholding (box 17 of your W-2) refunded.

See https://ttlc.intuit.com/questions/2895760-which-states-have-reciprocal-agreements