UPDATE: I recently had issues with Air and Twhirl again, and thanks to @smmehadi I found there was two steps I didn’t do to *completely* get rid of all traces of Air. I added that step into the list below. For reference, here’s the official Adobe Air Linux troubleshooting page.

Adobe recently released the final version of their Adobe Air 1.5 desktop web application framework for Linux. It was a great day, and a lot of people immediately upgraded. Problems occurred immediately when people tried to run or upgrade their Air applications, and found Air was complaining about damaged applications or .air files.
I had the same, I couldn’t install the latest version of Twhirl, my number one choice for Twitter client. After several attempts and to-ing and fro-ing on Google and reading forums and even bitching about it on Twitter, I finally managed to get Twhirl running on Ubuntu, using Adobe Air 1.5
Here’s how I did it using Ubuntu (YMMV):

• Firstly, to make 100% that you’re going to succeed here, make peace with the fact that you’re going to have to reconfigure whatever Air applications you had.
• Update your system’s repositories to the latest by running “sudo apt-get update” and then “sudo apt-get dist-upgrade”
• Uninstall Adobe Air. There’s an uninstall option on your Applications menu.
• Delete any Air applications you may have installed, they usually reside in /opt/ on the disk.
• Delete your ~/.appdata/ folder
• Delete your ~/.adobe/AIR folder – NEW STEP
• Delete your ~/.macromedia/Flash_Player/www.macromedia.com/bin/air* folders – NEW STEP
• Make sure there is no /opt/Adobe/ folder, delete it if it’s there.
• Run “sudo apt-get autoremove adobeair1.0″ in a Terminal window.
  This will likely moan about packages having unmet dependencies. This is fine.
• Run “sudo apt-get -f install”, this will list the Adobe Air applications you remove in a previous step here, press Enter to allow Apt to clean up the files.
• Your system should now be suitably void of all Adobe Air traces.
• Head to http://get.adobe.com/air/ and download the AdobeAIRInstaller.bin file.
• In a Terminal window, make the file executable by running “chmod +x AdobeAIRInstaller.bin” in a Terminal window
• Run the file with “./AdobeAIRInstaller.bin” or double-click the file in your Nautilus/File Browser window
  This will install the base Adobe Air 1.5 framework
• After Air is finished installing, you need to make sure you have the latest Adobe Flash plugin installed. In a Terminal window, run this “sudo apt-get install flashplugin-nonfree”
• You are now ready to install your favourite Air applications. I installed Twhirl.
• Head to www.twhirl.org – Notice that, as of writing, the Twhirl guys made a special Linux build of the Twhirl client (Currently it is here http://www.twhirl.org/files/twhirl-0.8.7-air11.air)
• Allow Air to Open and install the application

And that’s it, it might seem like many steps, but really, it’s just a simple way of cleansing your system from any unwanted Air leftovers of previous installations, and installing the latest version of Air and your favourite Air applications.
It’s worth noting too, that this seems to only affect currently installed Air applications. Any new applications that I installed after upgrading to Air 1.5 seemed to work just fine, so it seemed to have something to do with either leftover config files, or application files, or possibly the way Adobe Air interfaces with the Apt package management setup. I have a suspicion it’s the latter, but I don’t know the system well enough to provide proof. But at least what I did above worked.