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I too am new to sonic birth. I installed and run it under Lion with no problems at all. When it installed, did it ask you to enter an administrative account password? Is your user account you are installing from an admin account? If not, change it in system preferences under Users & Groups, or when prompted for a username / password when installing, change the username to an admin user and enter an admin password.
If you do this and you're still having trouble, open a terminal window (Applications > Utilities > Terminal) and do the following
I think your problem is likely to be permissions-based. If it's still continuing, perhaps it is missing a necessary component. You should open the console (Applications > Utilities > Console), then launch SonicBirth and look in the console to see if you can find what's keeping SonicBirth from launching.