There are some important things that SQLite does not do. It is not client/server; a process must be able to fopen() the database file. NFS and SMB are options that can convey access to remote systems, but performance will not likely be good. Only a single process can write to the database at any time; it does not support concurrent writers. The backup tools do not support point-in-time recovery to a specific past tim…
And the safety of your data depends on the quality of your network filesystem's locking implementation. It's not too difficult to design a locking method that works most of the time, but it's a lot harder to build something that guarantees mutual exclusion over an imperfect network.
On a single machine, file locking is generally reliable (mediated by the kernel). Multiple clients can access the same database simultaneously, with readers and writers taking turns, and all that happens is you pay the performance penalty of serialized access.
But if your client thinks it has a lock on a remotely-stored database file, but the server-side lock has actually expired and another client is writing to the database, your data is likely to get hosed.