When you use the phrase "the others," you are basically saying "all the other things." For instance, if I say "Two of the apples were good; the others were rotten," that means that of the apples I had, *all* but two were rotten.
So let's look at your first alternate:
"The stamps in this section are from the United States. The others are from the other places in the world."
This would mean two things: 1) All the stamps outside of that section are
not from the United States; 2) The other stamps are from
all the other places in the world. Here, using "other" without "the" is much better. Now it just means that
some of the stamps are from
some other places in the world.
Now for the next one:
"You can buy this shirt and the other one. Which others would you like?"
These two sentences are contradictory. By saying "the other one," the first sentence tells me that there is
one specific shirt that I may buy in addition to "this shirt." "Another" means "one more," while "the other" refers to one specific other thing (with the implication that I know what that one other is).
Now, if you follow this with "Which others would you like?" you are asking which shirts (plural) I would like. Since you have just told me I may only have two specific shirts ("this one" and "the other one"), this question doesn't make sense. If you ask me "Which others do you like?" I might surmise that you are asking my opinion on several shirts, even though I might only get one. But if you ask "Which others would you like?" it sounds like you are letting me pick several shirts.
By the way, I wouldn't say "Which other would you like?" In a new sentence, I'd prefer to see a noun or pronoun after "other," so I'd say "Which other one would you like?" However, I don't think that kind of small distinction would be tested on the GMAT.