Can I put the database on a network drive?
Yes and no. You should not move just the database. Instead, you should install a separate copy of MemberTies on the networked computer, and place all the databases you use on that computer in the normal location (see the Server and Database Overview for more information).
Once installed, you can use the Server Utility to set up the server to run as a Windows Service. That will ensure that it is always up and running.
Then on each networked machine, create a new database profile that points to the IP Address of the server machine.