The rebate is taxable to the extent you received a previous tax benefit from any deduction related to property taxes.
For the 50% of your home that is a rental, $1500 should be treated as rental income for 2023. (You could also reduce your current year property tax deduction by $1500, it has the same effect.)
For the 50% of your home that is your personal property, you have to look at your 2019 and 2020 tax returns to determine if you received a tax benefit from deducting property taxes. If you used the standard deduction in 2019 and 2020, then you did not receive a tax benefit, and the $750 rebate per year that is allocated toward your personal property is not taxable. If you did claim the itemized deductions on schedule A and included property taxes, then the fact that you deducted the taxes in 2019 or 2020 (or both) means that the rebate is taxable now as a taxable recovery, meaning a reimbursement of a previous deduction. Taxable recoveries are entered as miscellaneous income in the "Other uncommon income" section of the program.
