Does "Day of the Tentacle: Remastered" include the Maniac Mansion easter egg?

The original version of Day of the Tentacle released in 1993 included an easter egg: The complete predecessor Maniac Mansion was playable by interacting with a computer in the game.

Does the remastered version which was released in 2016 still include this?

Yes. The remastered release actually includes the higher-resolution "Enhanced" version of Maniac Mansion released in 1989 as opposed to the 1988 original version of Maniac Mansion included in the original Day of the Tentacle.

In the 2016 remaster, the sound for Maniac Mansion is emulated and sent through your operating system's default sound device, rather than using a PC Speaker like the 1993 original. This is likely due to many (/all?) modern PCs not having a PC Speaker.