There have been a number of threads lately about hacked websites where attackers are gaining control of the servers these sites are being hosted on. As such, these servers should be treated as completely compromised.
Often you utilize the server unknowingly until you discover the hack. Sometimes you have to connect to the compromised server to investigate.
What possible risks exist from utilizing a server in its compromised state from another machine?
I mean everything from visiting a compromised webpage on a site hosted on the server to SSH'ing directly into the server from another machine. What malicious actions can be run against the remote machine accessing the compromised server or what potentially sensitive information can be used to aid an attacker in then taking over your remote machine? (RSA keys for example, or utilizing an active SSH session)