Given that Arq, Time Machine, and Carbon Copy Cloner* use snapshot based backups, there is no need to close any databases.
I have DEVONthink open 24 hours and use all 3 backup apps mentioned. Since I use Arq with Google Cloud & Storj, that’s 4 backups per hour.
Never once has this been an issue. You can even modify content in an open database while backups are running. Snapshot based backups are frozen in time, it doesn’t matter if data changes while backing up.
(*when used with APFS volumes)
Also, see this comment for documentation screenshots from Arq and CCC that confirm this.