House prices would still increase and decrease as their local markets fluctuate. If there are many buyers or too few sellers, house prices can go up.

The difference is that house prices won't regularly go up year over year as they are now. Houses are acting as a cash-equivalent or savings mechanism. Just somewhere to park wealth. Without fiat inflation, that utility would gradually disappear and houses will fall in prices as demand for them decreases and then stabilize at their value for living in.