There are 2 methods to remove the DND notice from the lock screen:
Method 1:
Tap or press on the DND notice until it turns white, then tap on the X.
Method 2:
Swipe the DND notice to the left, then tap on Clear.
The notice will not appear again as long as DND remains continuously ON.
However with each method, the notice will return if DND is turned OFF and then turned ON again.
I don't want the DND notice to appear because I have a custom lock screen with my contact info in case the phone is lost. The DND notice covers up my contact info.
The methods above work well for me because I keep DND on almost continuously to avoid spam calls. In the DND settings under Allow Calls From, I have specified All Contacts. This allows calls to ring from anyone in my Contacts, but all other calls go to voicemail. I only turn DND off if I'm expecting a call from someone not in my Contacts, which rarely happens.