GitHub Updater now gives some feedback when the API responds with an error. What I’ve done is capture the HTTP error code and add that to an admin_head hook.
I’ve also included some data if the error is the result of banging against GitHub API’s rate limit for unauthenticated accesses.

Error Codes

As these are HTTP response codes it’s pretty simple to figure out. A 401 means that the authorization is incorrect.

• For Bitbucket repos this can mean not having read access to the repository or having an incorrect user/pass entered in the Settings.
• For GitHub repos it can mean using an Access Token for a public repository or having the incorrect Access Token for a private repository.

A 403 error usually means that you have surpassed GitHub API’s rate limit for unauthenticated access of 60 per hour or you have not included an Access Token for a private GitHub repository.
If you have knocked up against the rate limit, the error will tell you when your rate limit will be reset.