In my experience, this comes down to the difference between acronyms and initialisms. Acronyms are pronounced as a words, while initialisms have each of their letters pronounced individually.
I would expect to hear the FBI, because FBI is an initialism (aloud: "eff-bee-eye"). Whenever its full form (Federal Bureau of Investigation) is normally preceded by "the," I would expect its initialized form to be as well. For organizations that do not normally have a leading "the," I probably would not expect one for its initialized form, either, but I cannot think of an example.
I would not expect to hear the NATO, even when I would expect to hear the full form as "the North Atlantic Treaty Organization", because NATO is an acronym (aloud: "nay-toh", not "enn-ay-tee-oh").
Of course, I would expect the NATO member when NATO is used an adjective to describe the member of NATO.
I have never heard a formal rule for this.