Do I need to stop the service to back up the database?
In SQL Server you do not backup the database's files directly.  You issue a BACKUP command which creates a new file, usually with a .bak extension.  You do not need to stop SQL Server to do this.  It is this new file which is moved to tape &amp; off site.  To restore you will present this .bak file to a RESTORE command.  Here's a previous question on the subject and the Microsoft site.  I imagine the other RDBMSs would work similarly but I claim no expertise with them.  Read the publisher's documentation.

For a flat file system I imagine you would definitely have to stop all write activity to ensure integrity of the data within the files you may hope to restore later.  Again, ask the developers.