I am paying money for my primary residence on a land contract to the seller. Can I claim that monthly payment as an expense similar to rent?

Rent may be deductible on your state return, if your state has such a credit (some do; some don't).   If your state is one of them, we'll ask rent-related questions when you go through your state interview.

But not on your federal return, rent is not deductible.   The IRS won't let you deduct rent for your personal residence.  However, there's a deduction for home offices, and expenses related to rental property you own are deductible.   Don't enter anything in the Rentals and Royalties section of TurboTax unless you rent out property you own to others for rent income.   If you started entering a rental property (Schedule E) on your federal return, but should not have, please delete it. 

How to delete forms in TurboTax Online

How to delete forms in TurboTax Desktop

Is Rent Deductible?