This feature got added a long time ago and I use it all the time.  For some reason it stopped working recently.  Maybe it's some funky combo of my Linux + xmonad combo, but in inspecting the inspector, it seems like the 'click' handler that'd close the tab isn't being triggered any more.

I've looked through the git log of the relevant file:
https://chromium.googlesource.com/chromium/src/+log/master/third_party/WebKit/Source/devtools/front_end/ui/TabbedPane.js

but didn't see anything related.