Any time more money is injected into an area, it creates inflation. For example, when women started to enter the work force starting in the 70/80s, it lead to a dramatic rise in prices. Of course, I'm not saying that women entering the workforce is bad, I'm saying that when households effectively started doubling their income, it didn't result in more saved money, it resulted in higher house prices.
The same goes with the income tax deduction for mortgage interest. The idea is that more money would go back to the mortgage payer, however, all it did is factor into the calculus of buying homes, so people could funnel that "savings" into a higher mortgage payment.
If the US decided to give everyone $1000/month for free, I'm pretty sure that would just make the house prices higher by the equivalent amount.