Edit: After half day, my wife’s Mac started having the same problem. I have then applied the solution that I found in the Apple
OS X Mountain Lion Airplay Feature Does Not Yet Support All Macs
Yesterday we seen Apple unexpectedly announce OS X Mountain Lion. One of the main features a lot of people got excited about was Airplay
Your Mac is on TV. This new feature in Mountain Lion is made for an audience. With AirPlay Mirroring, you can stream what’s on your Mac to your HDTV via Apple TV. Show web pages and videos to friends on the couch. Share lessons with a classroom. Present to a conference room. It’s a big deal for your Mac. And for everyone around it.
Today when I tried to make a video about Airplay with in OS X Mountain Lion I discovered that I did not have any Airplay options. I am using an Late 2009 iMac 3.06ghz but 9to5mac also reported that the option was not there on their 2.4 GHz 2010 Mac mini or 2010 MacBook Pro with a 2.66 GHz processor but it was working on their 1.8 GHz MacBook Air. This could suggest that this feature is just going to be for I5 and I7 Machines and Apple could be restricting the features for dual core Macs
Let us know what Mac you are using and if you have the Airplay options in OS X Lion
OS X Mountain Lion