Should I play Maniac Mansion before Day of the Tentacle?

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.

I am going to give another reason for No...

Maniac Mansion was released in 1987. It was a pioneer in the graphic adventures genre, setting a template for future generations of graphic adventures. The standard for adventure games (whether with or without any form of graphics) was to type in your actions. You needed to type in the actions correctly or the game would not progress. Maniac Mansion gave you the ability to navigate with a pointer, select on a verb and an object. You no longer had the mistake of mistyping. At the time, Maniac Mansion would have been seen as a massive improvement on the adventure genre.

Maniac Mansion was released on a variety of platforms, but in its greatest form it was with CGA graphics. The sound is also made up of primitive beeps and no speech support. The game also contains numerous dead-ends, and depending on the copy you are using (and how you emulate the game) potential for needing the copy protection documentation.

Day of the Tentacle was released in 1993 (6 years after the first game). The game builds upon the same game engine but there has been a lot of work between the 2 releases (at least 6 games building on the same engine were released between 1987 and 1993). The engine had gone through major changes.

Day of the Tentacle had better mouse support, had better design (no dead ends etc), had voice support (CD version only, although the very start had voice in the floppy version I believe) and better graphics (greater range of colours giving it the comic book/cartoon style).

Of course, games have evolved since - with each iteration having better graphics, sound and user interfaces. This makes later games easier to play (which is true for any genre).

Now, if you have played Day of the Tentacle or similar games of this standard, and then go back to playing the first game afterwards you may be in for a painful experience. Attempting to adjust to the more primitive interface, graphics and sounds may not allow you to enjoy the original game.

All of this depends on your general experience with games and your own tolerances. However, it is feasible that if you play Maniac Mansion after Day of the Tentacle you will not enjoy the game as much. On the same note, if you have a low tolerance for games designed on a C64 then you probably won't enjoy Maniac Mansion either way.

As others have stated, you can play Maniac Mansion within the Day of the Tentacle. This is the original port from the C64. An "enhanced" version was released later on (improvements in the graphics). A mod team did convert the game into a 1990-style release (Maniac Mansion Deluxe Edition) although there are differences to help combat the game dead-end scenarios.

If you play the original or enhanced version it would be better to use the ScummVM emulator as it gives you several features from other LucasArts games (period key to skip dialogue, better mouse support etc).