I haven't faced this on Heroku, but elsewhere. My suggestions -

1. As a first step, see the server logs in case there is any startup error.

2. Increase server timeout value. Check if Heroku supports this.

3. Check why application startup is taking more than 60 seconds. Are you doing some complex DB operation on startup, some network call etc. ? Keep startup process lean.

4. If everything else fails, migrate to a new platform. AWS would be my recommendation.