When does its take an apostrophe?

It's with an apostrophe is a contraction of it is or it has. Its without an apostrophe is the possessive. The test is to expand the contraction: if it is or it has fits the sentence, use the apostrophe, and if not, do not. The dog wagged its tail expands to the dog wagged it is tail, which fails, so no apostrophe. It's raining expands correctly, so the apostrophe belongs.

The reason this error is so persistent is that it violates the pattern everything else follows. English marks possession with apostrophe-s almost universally, the dog's bowl, Sarah's car, so writers reasonably extend that to its and get punished for it. But possessive pronouns as a class take no apostrophe, his, hers, ours, yours, theirs, whose, and its belongs to that group rather than to the noun pattern. Note that hers and theirs have no apostrophe either, and that her's and their's are simply always wrong, which is a useful consistency check. Spellcheck will not catch a wrong choice here since both forms are valid words, which is why this error appears in published books, corporate signage, and expensively printed materials with some regularity.