If I were to be evicted, would my landlord have any entitlement to my income?

Your landlord would have to go to court (in addition to the separate eviction proceeding) and show that they are still owed money from you. If they successfully make their claim, then the court will issue a judgement against you. On the basis of the judgment your landlord (now a creditor) would be able to proceed against your assets. And your income is an asset. They could have a portion of your wages garnished (as set forth by state law). Additionally, they could attach your bank accounts and other personal  (non-exempt) property to satisfy the judgment.

Note: Many judgments are good for 10 years or more and many can be renewed.