It looks like the Ethereum classic vs. non-classic confrontation has acquired a new dimension.
Any Ethereum (non-classic) holder willing to take advantage of Ethereum Classic tokens (by selling them or buying anything for them) risks losing a corresponding amount of ETH on the non-classic chain. These are called replaying attacks.
How is it possible? As private keys on both chains are the same, any transaction broadcast on Ethereum Classic can be re-broadcast by a malicious actor on the Ethereum non-classic chain. As a result, when you transfer your ETC to somebody else and you are unlucky, your ETH might move as well.
The simplest worry is that you're going to get replayed in one chain, whenever you do something in the other: any strictly increasing subset of your contract calls can be replayed on the other chain.
You sell the coins, withdraw Bitcoin, and are happy. Until.. you log back in to your client tracking the non-Classic fork and see that your coins are gone. Where did they go? Some ETH-Classic supporter has sent them on to the same address as Poloniex' ETH-Classic deposit address, but this time on the main chain, by simply replaying the transaction in the main chain.
A more comprehensive explanation can be found here:
http://vessenes.com/do-not-mess-with-eth-classic-it-will-f-you-up
UPDATE: Vitalik offers a solution which mitigates replay attacks:
https://blog.ethereum.org/2016/07/26/onward_from_the_hard_fork