How we cite our quotes: All quotations are from From Here to Eternity.
Quote #1
WARDEN: He'd strangle in his own spit if he didn't have me around to swab out his throat for him.
Sgt. Warden is referring to Captain Holmes: it's a (gross) colorful metaphor. As the movie goes on, Warden becomes less of a spit-swabber for Holmes and seems to identify his duty more with being a good soldier than with kowtowing to superiors who are violating army regulations.
Quote #2
MAZZIOLO: Maybe he won't get it. All he did was to get drunk. It's a soldier's nature. It's almost a sacred duty once in a while.
Mazziolo's talking about duty in another sense here. He's saying that the soldiers need to "let off steam" sometimes. It's a necessary pressure-valve for their real duty: defending the country.
Quote #3
LEVA: I can't issue live ammunition without a signed order.
HENDERSON: The captain ain't here, jerk!
LEVA: Sorry, no orders, no ammo.
This is an example of a soldier misinterpreting his duty as merely following orders. It's a state of emergency—the Japanese are bombing Pearl Harbor!—but he's still sticking to the orders he was given before that emergency broke out. Warden ends up forcing him to move aside so he can get at the ammo.