Using desktop computers to manage an SQLite database on a fileserver?
The issue with this is that as you scale SQLite, you run into the fundamental problem that it doesn't have row-level locks, only table locks. The "thing" you run out of first isn't a raw resource like CPU or IO or memory - it's that your app will get bogged down in lock management. You can have lots and lots of users sharing an SQLite database, but only your readers can tolerate waiting for your writers. If this database is going to be on a a shared drive, then that will only make your scalability problems worse as it introduces additional latency, making the lock acquire/release cycle longer!

Also, in a business, there is no such thing as "non-critical data" unless you are literally talking about people MP3 or recipe collections! If people use it to get their work done and for whatever reason it "goes away" then they will wonder why, and perhaps escalate it up their reporting line. I totally sympathize with how long it takes to get a "proper" database server (be aware that the DBA feels exactly the same way, when he's waiting for the procurement department to buy the servers!) but unless you are prepared to support something forever, don't deploy it. Or if you do, develop it with something like the free versions of Oracle or SQL Server, so that when the time comes, it can easily be migrated into the datacentre.