We see a couple of different puns spoken by the mechanicals. One pun is seen in Act 3, Scene 1 after Puck has changed Bottom's head into the head of a donkey. Bottom speaks one pun in the phrase "you see an ass-head" (III.i.109). This phrase has a double meaning. Figuratively, seeing an "ass-head" means seeing a "figment of your own imagination." However, it refers to a double meaning in that Snout is indeed looking at Bottom and indeed seeing an ass's head instead of Bottom's own head.
A second pun can be found in Quince's line, "Bless thee, Bottom, bless thee! Though art translated" (111). The word "translated" means to "change in form [or] nature," showing us that Quince is recognizing that Bottom has indeed been changed (Dictionary.com). However, "translated" also has a linguistic meaning to refer to one language being changed into another. Bottom is a weaver, and the name can refer to the bottom, or skein, that yarn is wound around. However, "bottom" can also refer to backside, and the slang term ass can also refer to backside. Hence, when Quince is referring to Bottom as being "translated" he is also pointing out that the meaning of Bottom's name has now clearly been changed to refer to backside, or the slang term ass.
In Act Five, Scene One of A Midsummer Night's Dream, Theseus asserts, "No die, but an ace for him," while watching the Pyramus and Thisbe play. This is a pun (although not necessarily one that translates well into modern English!) because "ace" used to be pronounced like "ass"; thus, Theseus is unknowingly playing off the fact that Nick Bottom--who is now playing Pyramus--had been turned into a donkey earlier in the text.
In Act Two, Scene Two, Lysander responds to Hermia's request to "Lie further off yet, do not lie so near," by riffing off of "lie": "For lying so, Hermia, I do not lie." This is a pun in that Hermia meant one definition of "lie" (to physically lay down somewhere), whereas Lysander is using the word to testify to his own faithfulness.