User blog comment:AYET/Dressrosa Devil Fruit Hunting/@comment-5780397-20150528110836

Keep in mind that Burgess does not know who has eaten what fruits. Probably he only knows about Sabo and Luffy, and possibly also Doflamingo. If he is not stupid, he has realized that Pica has eaten a Devil Fruit, but he was arguably keeping an eye on Luffy at the time, so he wouldn't know where to get it. However, it is also possible that he has already stolen it. The same goes for Sugar; however, this time he has no hints at all as where to search, because he has never seen the eater in action.

Now, I'm assuming that Blackbeard told Burgess not to confront Doflamingo, either because the latter is too strong/elusive, or because the Ito Ito would not really suit Burgess's fighting style. Attacking Isshou would cause another war to burst, and anyway, Burgess probably does not stand a chance against him, and he knows it.

As for Luffy, we have seen that the Gomu Gomu is incredibly useful in melee combat. Think what Burgess could do with it, considering that a good portion of Luffy's strength comes from that fruit. However, we are assuming that Burgess wants to kill Luffy only to steal his fruit: in reality, he might have been ordered to do so by Blackbeard, since Luffy views the Yonko as his enemy, and "while he is at it", also to steal his DF.

1- Mera Mera no Mi: a Logia is still a Logia, plus we've seen that Burgess's Haki is not the strongest one around. He'd also get powerful ranged attacks, while resorting to his brawn in close quarters combat.

2- Gomu Gomu no Mi: excellent in melee, and grants resistance to certain types of attacks. Plus, he kills a possible enemy of his crew.

3- Ishi Ishi no Mi: my only objection is that Jesus Burgess uses strength has his primary weapon, and there is no guarantee that becoming a huge statue would make him stronger than he is. I mean, Pica was so dangerous because of his weight, but Jesus's brawn might rival it.