The simple answer is no, DigitalOcean does not require you to turn off your droplet to take a snapshot or backup.
In the past, DigitalOcean did require the server to be shut down first. You used to have to shut down, take a snapshot and reboot. This is a process anyone running a server does not want to do, luckily now you can keep them running.
DigitalOcean however still recommend servers are powered down for consistency, saying that in the millions of backups we have performed for customers, we have not seen an issue. If you are really worried about fast transacting databases, for example, we have a range of database backup solutions targeted at all the major providers
At SnapShooter we do not turn on or off servers so we will take a snapshot in the current state your server is in. You may find a snapshot takes a little bit longer to take a snapshot while online. Snapshots also take longer if your server is under heavy load. From experience, we don't notice the server's performance drop while taking snapshots.
How long do DigitalOcean snapshots take?