Blackbeard did not send Burgess to kill Luffy, he did not even know Luffy was at Dressrosa to begin with. Blackbeard originally had send out Burgess to Dressrosa to get the Fire Fruit. Later it was Sabo who ate the Fire Fruit, but it looks like not much people knew about this. The only thing most people really knew, was that Lucy had eaten the Fire Fruit.
Later when Sabo blew up the colosseum, Burgess realized that Lucy was not Luffy any more, because Luffy already ate a Devil Fruit before and could not eat a second. He just did not know who actually ate the fruit and thought that person would have been long gone by now. That's why he went for the next best thing, not having to go home empty handed. Though, when Sabo showed Burgess, that he had eaten the Fire Fruit, Burgess completely forgot about Luffy and turned his attacks towards Sabo instead, still hoping to get the Fire Fruit eventually.
On a side note: Blackbeard never targeted Luffy for personal reasons. He originally wanted to kill Luffy to make a name for himself, but after he became a Warlord, he did not care much for Luffy any more. He could have easily destroyed Luffy at Impel Down, but he had other ambitions by that time, other than the one he had at Mocktown, Jaya.