Its is a possessive pronoun, like his, hers, ours, yours, mine, theirs... no-one would apostrophise any of them. It's just that people get confused because there are two 'its'..
An apostrophe signifies a missing letter or letters. It's thought that nouns used to have a 'possesive' case e.g boy's would be 'boyis', the apostrophe shows that the 'i' is missing.
There used to be lots of 'short cuts' in old English orthography; such as X represented the word 'Christ', so in old parish registers you'll find Christopher written as Xtopher.
I did a bit of paleography many years ago, it's fascinating and does make you realise some things which had always been a puzzle! However, I didn't use it very much so can't remember all the 'shortcut' symbols now...