No, it's not possible. With the exception of Script commands, nothing in the blockchain is executed by Bitcoin Core: it's just data. Scripts are only able to do things like hashing and arithmetic, so it's hard to imagine any sort of Script-based malware.
It's conceivable that malware (once delivered to your computer via other means) could use the blockchain to download additional malware or transmit signals to infected machines, but it seems like it'd be way more trouble than it's worth to me. (if you have any sort of Bitcoin-informed malware, the most profitable thing to do would almost certainly be to steal the user's bitcoins, not use the blockchain for clandestine messages)