If you let anonymous users vote for
any post on a site just one time and
you log that vote by the user's IP,
what's the likelihood that you'd be
banning other users from voting
Unless that page is extremely popular, it's very unlikely that someone else being assigned the same IP address by the ISP would also visit it.
Edit: Users using the same IP address due to NAT are a much bigger problem and probably a deal-breaker for using the IP address. I'd be less worried about corporate networks than about private home networks: very common, and having two people in the same household wanting to visit and vote on the same site is rather more likely than two random strangers.
and that the original user would be able to vote again after a certain amount of time
because their IP address has changed? I'm guessing almost certainly.
It's not just a question of time; most ISPs assign IP addresses upon connect, so all someone has to do to get a new one is to reinitialize their DSL connection (or whatever they use).
Does this mean there is no reliable way of tracking anonymous users indefinitely?
Correct.