Is there any way to set CodaLab so that one's get email when a new result is posted on CodaLab in a given competition?
This isn't currently possible with the free hosted version of CodaLab Competitions. However, you can always download the CodaLab Competitions source code yourself, add the feature, and deploy the modified version on your own web server.
Of course, changing the CodaLab Competitions source code makes sense only if you're the one running the competition. If you're a participant in someone else's competition, there's nothing you can do other than write a script that periodically fetches the results page, compares it against the previously fetched version, and e-mails you if it notices a new entry.