There are many things about Martin that irritate Douglas, but the one thing that infuriates him more than anything else is seeing him be his own worst enemy.
Arthur is Arthur, and Douglas makes unabashed use of his Arthurishness when it suits him - such as when it brings him a bottle of Talisker, or when the flight is unbearably boring, or - well, when the opportunity presents itself. Douglas makes fun of Arthur knowing that most stuff will just slide past him anyway, and Arthur will keep on being the same perpetually cheerful dolt. He never says anything truly hurtful to him, because that wouldn't be funny. He doesn't expect anything more from Arthur.
Martin, however, could do better. He should take Douglas' ribbing as a compliment, really, even though Douglas knows Martin would never see the (admittedly, somewhat perverse) logic in that.
Douglas will never tell Martin, but he's reluctantly impressed with how he made himself a pilot. He is, however, entirely underwhelmed with how Martin's evolution abruptly stopped there and refused to budge. Martin has made his entire world about Being The Captain, and that's why it's so easy to tip him off-balance and survey the painfully hilarious results. It's also why Douglas would rather play a game of charades with Arthur than praise Martin about any aspect of his captaincy, because according to the Martinish laws of physics, any perceived successful action seems to cause an immediate and catastrophic opposite reaction.
Douglas still has nightmares about their deplorable encounter with Boston airport security; in some of those nightmares, Martin ends up embroiled in a Kafkian maze of processes; in others he ends up shot. Douglas wakes up with his heart beating a little faster than medically advisable, and absolutely furious with Martin for making him feel like that.
There are many things about Martin that irritate Douglas, but, perhaps, the thing that infuriates him the most is that he cares enough to be infuriated about the things Martin does.