An argument was posited above that this spelling convention (which, in my opinion, is largely arbitrary, simply to distinguish the genitive marking of it from the contraction of it is) is logically explained by an analogy between its and other pronouns’ genitive (i.e., “possessive”) markings.
Note, though, that there are THREE senses of its with genitive marking which we may consider, parallel to my/her vs. mine/hers vs. me’s/her’s (yes, this exists, via phrasal genitives; you’ll see).
The sense parallel to my/her is most obvious; “My/Her/The dog’s/Its name is rather long”.
The sense parallel to mine/hers is not so hard either: “If you only remember one name tonight, remember mine/hers/the dog’s/its.”
Finally, we can also consider a phrase like “The woman standing next to me’s dress”, as used to refer to a dress belonging to the woman who is standing next to me. This is indeed perfectly grammatical, and it would not at all do to substitute my for me’s (without switching parsing and thus changing the meaning; and substituting mine would be even worse, being ungrammatical on any parsing). So there are situations in which one has me’s, and him’s, and her’s, and all the rest of it. “The woman standing next to me’s/her’s/the dog’s dress”. Just as well, we can substitute it. “The woman standing next to it(’)s dress”. In this, with the meaning of the dress belonging to “the woman standing next to it”, would one write its or it’s? If still the former, then this would entirely be unaccounted for by any putative analogy between “its” and “his”. (However, I actually would, in my own writing, go for the latter in this situation, I think)
At any rate, any argument purporting to give a logical reason for why its should be spelt with an apostrophe has to cover not just the first case, but the second case as well, as well as the third case (either by explaining why the third case should also be spelt with an apostrophe or why the third case is different enough to not need to be spelt with an apostrophe, depending on position taken).