Will I mess up my ForeFlight account if I change my Apple ID?

Changing your Apple ID information does not effect your use of the ForeFlight app.  The use of the app is predicated on your signing in with your ForeFlight email and password.  

Your Apple ID, in terms of ForeFlight, is only used to download the ForeFlight app from the Apple App Store. It does not attach any of your ForeFlight account information to the app.

Your ForeFlight Login is what ties your account to the ForeFlight app. You can log into any ForeFlight app no matter what Apple ID was used to install the app and expect to have your synced items brought into the app.