They'd be renting to criminals. Aiding criminals is also illegal. Might as well go to whatever country they came from and take everyone out of their prisons while we're at it.
Most of them commit more crimes later on anyway. Most don't pay taxers and send money out of our economy. If they have no where to live, maybe they'll go home.
There are multiple reasons why renting to illegal immigrants should be illegal. It should be required for all landlords to be aware of who they are renting property to. This is not to say that all illegal immigrants are bad people, but the point remains that if a landlord does not even know the citizenship status of their tenants, they are less likely to be diligent about keeping violence and drug activity out of the neighborhood. If they are willing to rent to illegal immigrants, they are likely to allow other illegal activities to go on at their properties, which can quickly destroy a neighborhood.
Renting to illegal immigrants should not be illegal. When immigrants come to the country, they are just coming here for a better life. When you arrive here, it is extremely hard to become a citizen, because of all the obstacles and other red tape they place upon you, so it could be a while before you become a citizen, so everyone should have a right to rent.