Ableton announced on their Answers site that a bug has been found when running Live on OS X Lion and they do not recommend upgrading at this time.

Update(8/22): Ableton Releases Live 8.2.5 with OS X Lion Support

(last updated 13 July, 2011)
Live 8 is being tested with OS X Lion. Here’s the current state of affairs:

We have found an issue that causes Live to freeze when sending MIDI out, either to external hardware or internally via the IAC bus. We are currently working on a solution for this problem, which we will release in a bugfix update as soon as possible. Until this update is available, we do not recommend updating to Lion.

Note: In Lion, Apple has hidden the /Users/Library folder, which contains Live’s user preferences and library. This should not affect most users, but in certain cases you may need to access these folders directly.

To make /Users/Library visible, hold down the “Option/Alt” key, and choose “Go to Library” in the Go menu of the Finder.

If you are currently testing Lion and find any additional problems with Live, please let our technical support team know: