So for housing it depends on the local laws (1), but for labor this has actually been heavily studied, and the general consensus (2) is actually higher demand for native-born labor. All of those immigrants need goods and services, after all, so they increase demand. Focusing on immigrants providing labor without mentioning that they buy things as well is assuming your conclusion.
And the demand is not just for doctors and other highly credentialed labor. The most nativist work on the Mariel Boatlift (3) (by Borjas) only found a decrease in wages for native-born high-school dropouts, and the general consensus of other economists seems to be that his work was faulty. The general consensus on Mariel (cite fn2 again) is that it was good, economically, for the native-born people in the Miami labor force when an additional 7% of the labor force suddenly arrived as immigrants over the course of a few months.
1: How much housing can be built legally is the key factor here. Immigrants can drive up the prices if no more housing can be built but that's just another way of saying that "no more housing can be built" is a really terrible policy that is enormously destructive. If building housing is relatively easy, then it doesn't drive up prices. In fact, immigrant labor is often used to build this new housing. Don't let your area become like the Bay Area and you'll be fine.
2: Wearing my physics hat, I am not comfortable saying that anything in economics is actually proven.