The phpBB install process, throughout the 3.0.x line, has an issue with validating the existence of an SQLite database file given to it. This bug is really easy to replicate, here's some step-by-step instructions.

1. Begin the install process as usual.
2. When you reach "Database information", select "SQLite" as the database connection method, then enter a non-existent filename in the "Database location/DSN" field. Attempt to proceed.
3. The install script will report a "Successful connection". Proceed as usual.
4. When you reach the point where the databases are written, many errors arise from attempting to write to the non-existent file.

For your convenience, I've supplied a fixed version of the offending file (functions_install.php) which you could integrate into the next version or whatever, if you want.