How to verify that a firewall closes inactive connections?

If it's available or can be installed, take a look at netcat. You could do something like this.

On the server run:

nc -l 31415

On the client run:

nc -w 5400 <server> 31415

You can change the port number to anything you like, just make sure you can reach it from where you are testing.

A timeout of 90 minutes (-w 5400) is set in the example above. Change that as necessary.

You can test that from multiple locations: On the server itself, on another server/device on the same network, on clients on the other side of any VPN, routers, or firewalls.

More helpful information about netcat:

http://www.thegeekstuff.com/2012/04/nc-command-examples/