The version with it is the one most style guides will probably recommend. The other version is also acceptable, but it is probably considered less formal and less traditional by most. You will probably hear that one more often in casual speech.
To analyse the sentence, it helps if we transform the relative clause into a main clause and see how it works with and without it:
- (?) His job is to clean windows.
- It is his job to clean windows.
- ? To clean windows is his job.
In the first sentence, his job is subject, because it comes before the verb. That works all right. It simply describes what his job entails, what he's probably doing every day. But for some reason I'd prefer the gerund here; the infinitive may be more acceptable in a longer sentence of this type; I'm not sure why:
1.a. His job is cleaning windows.
1.b. His job is to clean the windows of the Tower next Sunday, before the Queen arrives.
It may have something to do with balance inversion; to clean is only put after the verb here because of the long phrase that must come directly after it: otherwise, this phrase would go somewhere in the middle of the sentence, which would make it much harder to read.
Sentence 2 looks most idiomatic. I think the phrase "it is his job" is a fixed expression with a slightly different connotation: it is his duty to clean windows. You could say it puts focus on it is his job, presents it as the new and salient information—this as opposed to sentence 1, in which focus is on to clean windows.
The reason why this indicates different focus is that, in 2, to clean windows is subject, which is expressed by the place-holder subject it that points forward to to clean windows. In a regular sentence, the subject is topic (non-focus, known information) and the predicate has focus, so that fits nicely. In sentence 1, the predicate was is to clean windows, and indeed that is what the focus was on.
In sentence 3, to clean windows is subject; it is of the same type as sentence 2, except that the deferred subject is filled in at the place of the place-holder subject it. But then we are troubled by an entirely different problem: English usually doesn't like it when an infinitive with to is used as the (direct) subject. In this case, it doesn't look too good; that is exactly why place-holder subjects are used at all. We could use the gerund instead to get an acceptable sentence:
3.a. Cleaning windows is his job.
This would be just as fine as sentence 2, with similar focus on is his job.
Now let's return to the relative clause:
We have a guy whose job is to clean windows.
This would be of type 1 above, if you consider word order. It may be acceptable, but it doesn't sound as good as type 2 in formal language, at least not to me. This is probably for the same reason as concerning sentence 1 above.
We have a guy whose job it is to clean windows.
This is of type 2, and it sounds better. That is probably because it is his job is a fixed expression and focus is in the right place. It may not be entirely evident that what applies to a main clause carries over to a relative clause, but I think it is a fairly reasonable hypothesis.