There are a few downsides to changing the App ID, especially if the app was created a while ago. However, it's important to have control of your application, so these are just for reference:
- User IDs between old app and new app will be different (this is related to app_scoped User IDs, with newly registered Facebook apps)
- Permissions will need to be reviewed by Facebook again
- You will lose access to Graph API v1.0 if your app was created before April 30th 2014 (but this will be depreciated in April 2015).
- You will need to get your App Center listing approved again
Having made the above points, it's important to have control of the application. If you're not using a lot of extra permissions its recommended that you create a new App ID and update your application. If you have an existing database of users, their IDs will no longer correlate to the same user, so it's also best to start with a fresh Database, or find a way to migrate the data to the new User ID by matching Email addresses.