All application protocols today have very short timeouts, typically in the few tens of seconds range. Most TCP stacks have a 10 minute timeout by default, if I recall the value correctly - if a sent packet has not been ACKed within that range, the connection is aborted. DNS resolvers typically have a similar timeout.
And as for servers, while you're right of course that server ports don't get tied up, there is typically some memory cost to maintaining an application-level connection, even if idle. I don't know of any common HTTP server for example that, in its default settings, doesn't close connections after some short idle time, for this very reason. More sophisticated attacks, like SlowLoris, have been created for this very reason. A server which doesn't timeout idle connections can easily be DoSed by a handful of compromised clients.
And as for servers, while you're right of course that server ports don't get tied up, there is typically some memory cost to maintaining an application-level connection, even if idle. I don't know of any common HTTP server for example that, in its default settings, doesn't close connections after some short idle time, for this very reason. More sophisticated attacks, like SlowLoris, have been created for this very reason. A server which doesn't timeout idle connections can easily be DoSed by a handful of compromised clients.