It's definitely an interesting question. Increasing the pool of people able to buy a house increases demand, so that probably has some impact on the price.
Another factor is that people tend to think in terms of monthly payments, rather than overall price and interest rate. For example, I'm paying $1,500 in rent while saving $500/month for a mortgage. If the requirement for a down payment is removed, I may then think it makes more sense pay $2000/month for a house now with a low down payment than to keep trying to save a down payment while prices keep increasing.
FWIW, this is exactly what happened (with different numbers) when I bought my first house.
Another factor is that people tend to think in terms of monthly payments, rather than overall price and interest rate. For example, I'm paying $1,500 in rent while saving $500/month for a mortgage. If the requirement for a down payment is removed, I may then think it makes more sense pay $2000/month for a house now with a low down payment than to keep trying to save a down payment while prices keep increasing.
FWIW, this is exactly what happened (with different numbers) when I bought my first house.