The word "live" is not a singular form.
You have a concern regarding the dependent clause. I suspect that we can examine that concern even when the clause stands independently:
They're used as a place to live.
It may also be helpful to expand the contraction:
They are used as a place to live.
The subject is "they". The verb which agrees is "are".
The word "used" is also a verb, but it is part of the verb construction "are used". Since the "are" of "are used" agrees with its subject, the entire construction agrees with its subject. On its own, this "used" doesn't agree with anything.
The phrase "to live" is another verb, but it is not a part of the verb construction "are used". Instead, this infinitive phrase is traditionally regarded as a modifier of "a place". It's a non-finite form that has no subject and does not need to agree with anything.
The subject "they" and the compliment "place" do not agree. "They" is plural, but "place" is singular. There is no rule of English grammar that says that a complement must agree with its subject. Even though they share the same referent, they are separate references.
In this case, giving the subject and the complement the same grammatical number might be clearer and might count as better style:
They're used as places to live.
Still, this agreement is not required, and there are times when such an agreement isn't even sensible:
These cows are one herd. Those cows are another herd.
By the way, the word "live" is not always an infinitive form. It also happens to be one of the simple present-tense forms. Even in that case, it is not the third-person singular form. It agrees with first-person, second-person, and third-person plural subjects:
1st: I live. We live.
2nd: You live.
3rd singular: He lives. She lives. It lives.
3rd plural: They live.
The singular form is "lives".