Should I play Maniac Mansion before Day of the Tentacle?

While the other answers say a strong "no", I'll make a stand on this matter and say:

Yes. Or no. Depends on the exact meaning of your questions.

If it's along the lines of Do you need to read Hobbit before reading LotR to be able to understand the plot?, the answer is no. If it's Do you need to read Hobbit before reading LotR to be able to fully understand the plot?, then the answer is yes.

More specifically:

Are there any reasons for playing Maniac Mansion before Day of the Tentacle?

Yeah, a good one - understanding the background flavour (Edison family, whole Purple/Green Tentacle business, the Mansion itself, Bernard, the Hamster etc.) & all tongue-in-cheeks made about protagonists & antagonists from MM1 makes the game arguably a lot funnier, and playing adventure games is about fun after all,

Does anything occur in the second game that requires playing the first game to understand the plot?

Nothing important occurs that would require playing MM1 to be able to understand it - but a lot of completely unimportant things occur that require a knowledge of MM1 to actually be able to understand and enjoy them fully.

So it's basically down to tl;dr "do I need to read a prologue to understand a book?" - the answer depends on what you mean by "understand".

No, the narrative on Day of the Tentacle stands independent of Maniac Mansion. In fact, you can play Maniac Mansion from within Day of the Tentacle as the whole game is included as a hidden bonus.

http://www.eurogamer.net/articles/day-of-the-tentacle-retrospective

You find the hidden game by interacting with an old computer in Ed's room as Bernard.

Nope

The plot is perfectly meaningful without having played Maniac Mansion. There are some jokes (in-jokes?) that refer to Maniac Mansion, but other than that it's an independent story on its own.