When you are writing a program which uses a database driver, from Why is programming interface to database called driver?
There are databases that allow you to embed them within your process (like sqlite), but if you're working with a database that has its own process[es] (like postgresql), then yes, it(they) must be up and running in order to work against that database.
So some databases "allow you to embed them within your process (like sqlite)". If there is no DMBS running, to what does your process submit a SQL command, and what parses and executes the SQL command? (For example, in SQLite case.)
Does your application work, both when SQLite is running and when not?