Just for posterity, this is the command I ran:
./DaVinci_Resolve_16.2.7_Linux.run --nonroot -C ~/Programs/DaVinci_Resolve/
The following still happens:
Creating Common Data Dir : /var/BlackmagicDesign/DaVinci Resolve
mkdir: cannot create directory ‘/var/BlackmagicDesign’: Permission denied
It then crashed on the first few runs but then started. Cool tool, though!