It's is a known issue for use here: https://youtrack.jetbrains.com/issue/RSRP-450266
This bug is going to be fixed in the upcoming ReSharper 10.0.1 update.
As a workaround until the hot fix is available, avoid clicking on parent nodes in the tree. If you only click on tests, the output pane works as expected. Once you click on a parent node, the output pane gets "stuck", and stops updating. If this happens, right click on the tests you're interested in and select "Create new session". A new tab will open and you'll be able to run the tests again.

The workaround reveals another problem: the output window is not cleared upon running the test again. Any exception messages + stack trace remain visible and new output is appended.
A workaround is to click "Show Stack Trace in new window" the button in the unit test session window toolbar to the right. There is your output :-) But you have to click it for each test you want to see output.
@Mario Looks like another problem that you're referring to is known (https://youtrack.jetbrains.com/issue/RSRP-450273), and a fix is prepared for a bugfix update that we expect to roll out later this week.
Apologies.
The workaround (creating a new unit test session) worked for me too, but I had to revert back to 9.2 anyway due to http://resharper-support.jetbrains.com/hc/en-us/articles/213686757-VS-hangs-on-debugging-unit-tests
Hello @all! Please open ReSharper | Help | Check for updates and install R# 10.0.1. Our apologies for the inconvenience once again.
Thanks!