If I forget to close a file, a socket or any other resource in a Linux process, and the the process terminates, will those resources be freed?
Is there a difference if the process terminates normally or is killed?

SIGKILL won't give a change to clean as it will terminate right away. With SIGTERM, programs can catch it and create clean up processes. there's posix call for exiting and cleaning before the exit (man 3 exit man 2 exit)

What I was really asking was that if the death of a process that has not freed resources, can leave something lying around that will eventually bring the system to a standstill.

clive,
welcome to LQ. If a process terminates it usually frees the resources, so that other processes waiting for that resource can use it to execute themselves(this is known as CRITICAL SECTION when a process gets the resource and proceed towards the execution).

If you are talking about sockets, they are file descriptors. They are used by the application layer protocols to communicate with the transport layer protocols. You have to close them whenever the communication the is over.

In case you forgot to close sockets, they will remain open(this is the thing what we refer as port).