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 shutdown first. You used to have to shut down, take snapshot and reboot. Which is a process anyone running a server does not want to do, luckily now you can keep them running.
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 a longer if your server is under heavy load. From experience, we don't notice the servers performance drop while taking snapshots.