I have the same problem. This used to work fine at some point. I have a Nokia 7.1 with 8.1 Oreo. When I configure DAVdroid and then leave my phone alone for just a few minutes, synchronisation is disabled (both in the Android setttings and in DAVdroid). It seems that this problem is not exclusive to DAVdroid.
I think I may have found the solution while typing this message: There is a background activity manager in Android, you need to explicitly allow background activity for DAVdroid there (Settings->Battery->Background activity manager - sorry if the names are not correct, this is a reverse translation from my German Android UI to English).