I'm assuming you mean that someone in your family wants to pay a part of your mortgage for you which would be a gift and then they have it wrote off on their taxes as taxable income for them?
If your family member does pay off part of your mortgage then they can write that off on their taxes as a gift to you and they would get a benefit from the taxes.
If a family member does pay a mortgage principal for you then you should not be taxed on that amount because the money would not actually be going into your bank account.
Instead the money for the mortgage principal would be paid into the mortgage account so you should not be taxed on that money as income.
Some people are taxed on certain amounts of money that is gifted to them but if the money is paid directly to the mortgage and you don't let the IRS know about it then you should not have to pay income tax on it.