Can I access a database from a mapped drive on a share?
Short: You shouldn't.
Long: Firebird requires that databases are kept on a local hard disk. The reason is that the Firebird server needs to have exclusive access to database file and that is not possible on shares (mapped network drives via Windows networking or Samba). The exclusive access is needed since any attempt to write the database by two processes would really quick result in a corrupt database file.
So, keep your databases on a local disk. If you need to serve it from another computer, install Firebird there. It's lightweight and will only take few megabytes of your disk space.
Note: there is a configuration option in firebird.conf to work around this, but it is highly recommended that you don't use it.