Can I claim my childs off campus room expenses?

No.   Living expenses are never deductible by anyone.

No. Room & board (R&B), whether on campus or off campus is not a qualified expense for claiming an education deduction or credit.  R&B is also not a qualified expenses for scholarships to be tax free. 

There is one exception, R&B, even off campus (with limits) is a qualified expense for a distribution from a  Qualified Tuition Plan (QTP) (529 or ESA), to be tax free.

The only qualified education expenses are tuition, books, and lab fees. That's it, with no exceptions.

However, if ESA/529 funds are used, those funds can be used tax free to pay for room and board, provided that room and board is paid "in direct support" of the education. So if the student was not enrolled for the summer semester, room and board for that semester is not "in direct support" of the education and can not be claimed as tax free against the ESA/529 funds distributed.  There are other restrictions too.

If off-campus housing is utilized, and on-campus housing was available, the amount claimed for that off campus housing can not exceed what it would have cost to live on-campus.

For the "board" part, if the school offers a meal plan, then the amount claimed for that part can not exceed what it would cost if the student utilized the provided meal plan.

Generally, if the school has no on-campus housing available, the student will receive a letter informing them of that fact and advising they have no choice but to obtain "other housing". When you have a letter like that, then your claimed housing costs are not bound to the on-campus housing costs. However, the cost claimed must be reasonable. So no $5000/mo penthouse suite. 🙂