It's the same reason a lot of autists/lolcows don't wash - the feeling of water against their skin gives them sensory overload. An autist I knew said that taking a shower was like "being bombarded with hailstones", so he had baths instead, but others don't like being wet under any circumstances and don't wash at all.
I don't think Shane is autistic though - he doesn't show many of the signs, and besides, if he was (or even if he self-dx'd), he wouldn't shut up about it and would use it as a shield against criticism. He doesn't brush his teeth because he finds it boring, I suspect, and it doesn't give an instant reward. Shane has no impulse control as we have seen in his inability to stop himself lying and threatening people even though he gets busted over and over. Remember he is of low intelligence and has no self-awareness and a huge ego. He thinks he's better than anyone, and like Lucas Werner, his sense of superiority is so total and overwhelming that he thinks he's invincible and that his actions have no consequences. Therefore he thinks his teeth will never crumble, and when they started doing so he went into denial - remember him getting pissy at his dentist for daring to suggest he ate less sugar? This is just stupidity and ego, not autism.