What would be the difference between the below two sentences?
I want it to be completed by Sunday.
I want it to have been completed by Sunday.
What is the difference between the use of the present and the perfect infinitive here? Also, does the choice depend on the 'main' verb?
There is not much difference in the two sentences, practically speaking.
However, different tenses can affect when the sentences can be used. This is somewhat subtle, but to explore all the tense possibilities, here goes:
I want it to be completed by Sunday.
That sentence, using present tense (want) and past participle (completed) can be said only before Sunday.
In contrast, the "have been completed" phrasing adds a past reference to a present statement:
I want it to have been completed by Sunday.
That could be said before or after Sunday. As an example: "What? It isn't completed? I want it to have been completed by Sunday." (before now, in other words)
However, if you are speaking about the past, to be completely clear, the past perfect "had wanted" is precise:
"I had wanted it to be completed by Sunday."
That is someone saying that both the "wanting" and the expected "completion" are in the past.