Should I put my name on dad's building society book?

Thank you

This is too complicated for me to give a good answer, hopefully someone from over there will see this and be able to give you targeted advice.
There will be gov't agencies that specialize in elder care, also those that specialize in your type of low income situation so getting some advice there would be a good start. You should also get some advice from a lawyer about the possibility of getting POA, it is sometimes possible to have dementia yet still be deemed to have the capacity to make informed consent.

Cant get poa as hes not mentally capable of understanding

Yes iam in the uk .we work and claim tax credits to top up our wages im scared these are going to be effected.building society is a bank .you get a book instead of a card

I don't know what the building society book is, it sounds as though you are in the UK?
Regardless, if you get your dad to give you POA you can act for him without entangling his and your finances, a much better option.