You don't need to have played or know anything about Maniac Mansion to enjoy Day of the Tentacle. I would not even recommend to play MM first, because MM is stupendously hard and frustrating compared to DotT. Play it after.
BTW: In the version of DotT I played in the 90s there was a funny easter egg: A computer in the game was running a playable version of Maniac Mansion. I don't know if this is part of the remastered version though.
Remastered has it too. Steam version even has achievements for doing things that require you to get a small way through the original.
Agree on not needing to play Maniac Mansion first, too. I came to the original having no idea that MM even existed. The characters and story are great even if you don't know the references.
BTW: In the version of DotT I played in the 90s there was a funny easter egg: A computer in the game was running a playable version of Maniac Mansion. I don't know if this is part of the remastered version though.