sweepprivkey proposal from 2011 which was not accepted, that would have allowed one to instruct her bitcoind client to listen to transactions and sweep funds whenever they are transferred from one address to another.
There is also this question, "Is there a way to automatically send bitcoins from one wallet to another?", but it does not specifically ask about a service that would do that.
I was wondering whether someone developed, or is considering the development, of such a service. There would be security implications, as the service would have to know the private key of the address to sweep - is there any secure way to implement this otherwise?