Is it possible to forward bitcoins without the private keys? According to this question, there isn't an easy way to automatically send bitcoins from one address to another and I think that it might be useful.
So I thought about creating a website were people could upload their wallets and choose a destination address where the coins would be sent to. A bitcoin daemon would be constantly checking the balance and forwarding any received transactions. But then I was confronted with the problem of securing a collection of online wallets.

When a transaction occurs, what part of the message is signed by the owner? Since the destination address is always the same, is it possible to send the bitcoins without storing the private keys on the server?

Probably the whole transaction is signed (which means that I need the key because the input is unpredictable), but I don't know much about this so I might be missing something here.
Short answer: no.

Longer answer: to 'forward' coins one needs to spend them out of the first receiving address, and to the next one. in order to spend them, one needs to sign a transaction using the private key associated with the first receiving address.