mDNSResponder not logging man mDNSResponder suggests that sudo killall -USR1 mDNSResponder will enable warning and notice logging, and that sudo syslog -c mDNSResponder -d will enable log levels Emergency to Debug. Additionally sudo killall -INFO mDNSResponder should dump current state to /var/log/system.log.

None of these have any effect on macOS Sierra. None of the commands fail, but they don't actually log anything in /var/log/system.log either.
In my opinion parts of the log system of macOS 10.12.0 are flawed.

The output (dump) of sudo killall -INFO mDNSResponder is visible in the Devices > $hostname  "log" though.

First launch Console and choose your host device in the left pane. Enter a filter like mDNS in the search field at the top.

Then enter  sudo killall -INFO mDNSResponder in Terminal and watch the results fly past in Console.