From my experience (i) of wives, (ii) that the same garbage rarely makes its own way back into the house and thus needs taking out twice, it certainly means "Please don't again forget to take out the garbage [I say this as, upon [numerous] previous occasions, you have forgotten to take it out.]"
I find it fascinating that the three males who posted all interpreted "again" as relating to "forget," while the female poster sees it as relating to "take out." Perhaps men are more sensitive to being reminded about not forgetting?
In any case, I think we all agree that the sentence, absent context or other knowledge, is ambiguous. If it were "Please don't forget to sweep the porch again," where the porch has to be swept repeatedly because sand or dust blows onto it, it would be even more so.
I don't think there is any "real" problem with the placement other than that potentially created for people who can't keep track of the main verb of a sentence for the length of "to take out the garbage".
Don't forget eggs again.
Don't forget again eggs.
Don't forget to buy eggs again.
I don't agree with Paul Q in #5 or Packard in #7, *as to their reasoning,* though I agree the sentence is possibly ambiguous.
As Egmont pointed out in post #6, many tasks have to be done again; there is no mysterious return necessary, as Packard suggests, subsequently, in post #7.
If one normally takes out garbage, and does so, and there's a party later, then it may need to be taken out again; it's 'garbage' but not exactly the same items; as in Egmont's last example, for sweeping.
If I wanted "again" to modify "forget", I'd say it exactly as wanabee did, and I'd follow it with "You're always forgetting" or "You forgot yesterday." "Don't forget again to take out the garbage" sounds like robot talk.
In (my) real-world experience, the again would refer to forget -- it's a bit of chiding by the (female) speaker for having to repeat the same request time and again.
If you simply wanted someone to take out the trash, you would say, "Don't forget to take out the trash" -- there's no real need to use "again" to point out the obvious fact that it's a repeating chore.