Problem solved. It was the OpenTasks app.
SOLUTION: In the permissions settings of the OpenTasks app the option “Contacts” (in german “Kontakte”) has to be turned on.
Thanks for your help.
I use a custom view for People (Contacts) app to display just “My Contacts” group from DAVdroid. But after installing version 1.0 of DAVdroid from F-Droid, it overrides my settings and after some seconds “All other contacts” will be checked automatically.