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The FairTax has positive effects on residential real estate far beyond this narrow question.
Today’s homeowners, if they itemize (and 70% do not), pay their interest with post-Social Security/pre-income tax dollars. They then pay their principal with post-SS/post-income tax dollars. Those who do not itemize get no advantages at all. Under the FairTax, all homeowners make their entire house payment with pre-tax dollars.
With the FairTax, mortgage interest rates fall by about 25 percent (about 1.75 points) as bank overhead falls; this is a huge savings for consumers. For example, on a $150,000, thirty-year home mortgage at an interest rate of 7.00 percent, the monthly mortgage payment would be $999.12. On that same mortgage at a 5.25 percent interest rate, the monthly payment would be $830.01. Over 30 years, the 1.75-percent decrease in interest rates in this instance would result in a $60,879 cost savings to the consumer.
Finally, first-time buyers save for that down payment much faster, as savings are not taxed.
Under the FairTax, home ownership is a possibility for many who have never had that option under the income tax system. Lower interest rates, the repeal of the income tax, the repeal of all payroll taxes, and the rebate mean that people have more money to spend, and have an increased opportunity to become home owners.

You are forgetting that the buyer has to pay sales tax on the home and sales tax on the interest (it's a taxable service) above the federal longterm rate (currently 2.15%). That $150,000 home ends up costing $180,000 with tax (which would be financed) and with part of the payment on principal going to sales tax on the interest the total cost of the home $412,000 costing the buyer $52,000 more than it would otherwise. ~$64,000 of that is sales tax on the house (I didn't include the land, it assumed to be used) and on the interest.

Finally, first-time buyers save for that down payment much faster, as savings are not taxed.

Since most first-time buyers move into a used house, that makes for a zero tax purchase (not counting taxes on the bank/mortgage company service fees).

As I explained on another thread, that won't work. Most current homeowners refinanced (or bought originally) when interest was down near 5.5%. With current rates above 6% and rising, the effect of the change (by the time it had time to be enacted, and to percolate through the system) would merely restore the same old 5.5%-ish rate -- the only effect on the great majority of middle-class homeowners would be the loss of their biggest tax deduction.

The Democrats have wet dreams about the Republicans doing something that stupid to alienate their base.