as explained by Dr. Elrayah Eltahir Adam Khatir in the article “Towards Understanding Some Selected Puns from Shakespeare's Dramas.”

In this opening speech, Sampson and Gregory use the words colliers, collar, choler, and carry coals to point out the different meaning of the same word. Therefore these two are using puns in their conversation. Carry coals means to submit insults, colliers are workers in coal, in choler means to be angry, and a collar is a hangman's noose. Also, a pun is made with maidenheads, which is used as the heads of maidens and virginity. Naked weapon refers to an unsheathed sword and a male reproductive organ. To the wall refers to a sexual term as well its literal meaning. In this way Gregory and Sampson say the same word multiple times, but are using it in different contexts and with different meanings. (Khatir 1488)

Shakespeare doesn’t stop with a pun after one line but rather extends them over many lines and further increases their comedic relief.

“To carry coals” is an old English idiom meaning to be insulted, or to be the butt of jokes.  This stems from the status of coal carriers – or colliers, as Gregory makes mention of in the next line – in society.   It was a dirty and undesirable job, and so was deemed unfit for the better members of society.  For this reason colliers were looked down upon and often insulted, and hence the phrase.  So, when Samson says “Gregory, o’ my word, we’ll not carry coals,” he means that they will not stand to be treated like the basest members of society, i.e. insulted.  We can assume from this that from the very beginning the Capulets were roaming the streets gunning for a fight with the Montagues. 

The silly turns of phrase that Samson and Gregory bandy with relation to coal are good examples of puns, a type of wordplay that Shakespeare utilizes to great effect in his plays.  Here, Samson is using “to carry coals” in a figurative sense, and yet Gregory responds with the literal meaning of the phrase – if they were to carry coals, well then they would be colliers.  Samson then spins off the word collier, stating that “I mean, an we be in choler, we’ll draw.”  Here he plays off the similarity between the words collier and choler.  And Gregory then makes good use of the fact that choler and collar are homophones, and uses again Samson’s word draw when he says “Ay, while you live, draw your neck out o’ the collar.” 

The irony here of course is that Gregory offhandedly insults Samson throughout their entire exchange, even though the latter swore at the beginning that he would not tolerate it.  This further goes to show that their beef is purely with the Montagues; or at the very least that Samson is not very bright.