The song in the episode is sung by Chef, and it's titled "No One Sleeps," which is a direct translation of "Nessun Dorma." While the lyrics are altered, the melody is very clearly based on the original aria.
If you're looking for the exact lyrics from the episode, they're not readily available online. But you can easily find the original "Nessun Dorma" and see how it inspired Chef's hilarious version in the episode.