Same problem here!
Hardware:
Macbook Pro 13" Early 2011
Problem:
After upgrade to Mountain Lion I can't connect ANY external display or projector.
Thins that I've already tryed:
1) PRAM NVRAM reset - DIDN'T WORK
2) Fresh install Mountain Lion - DIDN'T WORK
3) Try others HDMI or VGA adapters - DIDN'T WORK
4) Downgrade (reinstall) Lion - all my adapters and external display worked again
Here is the chronology of the problem:
1) With Lion - Everything worked fine
2) Upgrade to Mountain Lion (10.8.0) - My HDMI adapter stoped work, but my VGA adapter worked fine
3) Update to Mountain Lion (10.8.1) - My VGA adapter died too. Now none of them work. No HDMI and no VGA.
4) Update to Mountain Lion (10.8.2 12C53 - I have an mac developer account) - DESPERATE TRY TO SEE IF APPLE FIXED, BUT NOTHING. SAME PROBLEMS, ANYTHING (HDMI and VGA) WORK.