Would this apply to businesses too, or just individuals? If businesses are not allowed to deduct interest expenses, that going to strongly constrain how much they will be willing to invest.

And if business ARE allowed to deduct interest expenses, but individuals not, then rich people will just have their company own their house, and rent it from themselves (or some similar adaptation).

In particular, for real estate, if loans become more expensive, fewer people will want to build housing, making the market even more constrained. (Now, I do see reasons for making lending more expensive, but that would be for financial stability reasons, not housing.)

To "solve" housing, I think the main approach should be to encourage the supply side. Make it easier and cheaper to build, and harder for local communities to enforce their NIMBY inclinations.

Within reason, though, because if take to the other extreme, you may end up with an oversupply of housing in many areas, which can lead to slums forming.