Several people are to blame for the fight between Mercutio and Tybalt. The first persons to blame are actually Lords Capulet and Montague for continuing an ancient family feud. Had the two families not been feuding then Tybalt would have had no reason to feel angered and insulted by Romeo's presence at the Capulet's ball. We especially see Tybalt interpret Romeo's presence as an insult when Tybalt speculates that Romeo has come just to make fun of their party, as we see in his lines:
What, dares the slave
Come hither, cover'd with an antic face,
To fleer and scorn at our solemnity? (I.v.57-59)
In this passage, the word "fleer" can be translated as "sneer," therefore the phrase "fleer and scorn" can be translated as to sneer, or laugh at, and show contempt for, showing us that Tybalt believes that Romeo has come to mock their party and show his hatred. It is the family feud as well as Tybalt's temper that has led Tybalt to feel insulted, showing us that Lords Capulet and Montague are two people responsible for the fight between Tybalt and Mercutio.
However, Tybalt is also significantly to blame for the fight as well. Had Tybalt not allowed himself to be governed by his temper and assumed false reasons for Romeo's presence, he would never have felt insulted in the first place. Tybalt even pointed out Romeo's presence to Lord Capulet who told Tybalt to let Romeo alone saying that Romeo is believed "to be a virtuous and well-governed youth," showing us that Lord Capulet did not feel insulted by Romeo's presence as Tybalt did (I.v.71). Therefore, Tybalt, with his fiery hot temper, is another person who is significantly to blame for his fight with Mercutio.
However, the third person to blame for the fight is actually Mercutio himself. Mercutio had absolutely no reason to be on the street that day and Benvolio begged him to go inside with him, warning that if they ran into any Capulets they would not "scape a brawl," meaning prevent a fight (III.i.3). Mercutio's reason to be out on the street that day was his own rash, impetuous decision. We may even be able to assume that Mercutio was out on the street that day in the hopes of starting a fight because he learned that Tybalt had challenged Romeo to a dual. Mercutio, himself, felt insulted by Tybalt's behavior and was hoping to fight Tybalt himself. This would explain why Mercutio was so insistent in starting the fight when Romeo backed down, trying to keep the peace, as we see when Mercutio cries out, "O calm, dishonourable, vile submission!" and challenges Tybalt himself.
Hence, many people are to blame for the fight between Mercutio and Tybalt, including Lords Capulet and Montague, Tybalt, and even Mercutio himself.
While Mercutio and Tybalt are both out looking for a fight in Act III scene i, the duel between them is ultimately Romeo's fault. Tybalt enters the scene asking for Romeo. Mercutio and Benvolio have not seen Romeo since the party the night before. Things become tense between Mercutio and Tybalt when Romeo suddenly enters the scene straight from his wedding to Juliet, presumably to tell Benvolio and Mercutio about his happiness and new bride. He stops short when he encounters Tybalt's rage. The audience understands that Romeo refuses to fight Tybalt because they are now related through Romeo's marriage to Juliet -- "Tybalt, the reason that I have to love thee doth much excuse the appertaining rage to such a greeting... I do protest, I never injured thee, but love thee better than thou canst devise, till thou shalt know the reason of my love: And so, good Capulet,--which name I tender as dearly as my own,--be satisfied." (This is a great example of dramatic irony.) Because Romeo refuses to fight Tybalt even after Tybalt hurls numerous insults at him, Mercutio feels like he must protect Romeo's honor and fight Tybalt for him. Had Romeo told Mercutio about his relationship with Juliet, maybe they all could have walked away from Tybalt until their new familial ties could be revealed.
Both here and elsewhere we find that some blame the author. As it seems that there is a source for everything that Shakespeare wrote, we might pardon him. Moreover, one might argue that he did recommend the customary virtues of justice, prudence, temperance, fortitude, faith, hope and charity. Perhaps Tybalt heard that Romeo kissed Juliet and later spoke with her the night before. He then may have thought that Romeo may be prepared to speak of how much he loves him all afternoon(see the Sonnets). Mercutio will not bear hearing much of that(see 2.3.137 or so). Therefore, perhaps Mercutio will insist on a duel. This then may have been Tybalt's plan and thus the two must share the blame.