Must I use past tense after "as if"/"as though"?
I just don't want to help him. It's not as if I had no money.
Google Books claims 17,300 instances of it's not as if I had, and 11,700 of it's not as if I have, which I think suggests that the short answer for OP is "both tenses are fairly common" in such constructions.
Note that the "refuted" statement ("I have money") can be expressed as "I am rich", where it's easier to see how the irrealis (not real, counterfactual) mood can be conveyed by a "subjunctive" verb form...
Even if I were rich I still wouldn't help
Note that in practice many native speakers today would use was rather than were there - but in neither case is there any suggestion of past. Also note this example...
...my throat feels swollen as if I had cancer in it; my guts feel as if I had cancer in them too.
...where clearly the speaker/writer is talking about an "unreal" situation now, not in the past.
In practice, most native speakers probably would use present tense in OP's specific example, if only to reflect the preceding I don't want to help. But there's nothing wrong with using the subjunctive, which in this case looks just the same as the past tense anyway.
There are many uses of as if in English. It can be used as a slang, comparison, conjunction, and exclamation etc.
The doubt you have is about using a present tense or past tense with as if. The past tense is common when you use as if for comparing things/people where comparison is not true.
He's behaving as if he were the king of the world.
In your example, it's used as conjunction where it'll mean like or as though. The past tense is not required.
I don't just want to help him. It's not as if I have no money.
My source for this answer is here. The article talks about the uses of as if in English.