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.