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.
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.
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.
LEVA: Army regulations state that I can't—
WARDEN: Are you blind? Give me the key!
LEVA: I got to obey orders, Top.
WARDEN: I'll see you get a medal!
Warden understands that his real duty goes beyond just following orders—if he's doing something to defend the soldiers and the island itself, the higher-ups presumably aren't going to mind.
ALMA: They'll find out you killed that soldier.
PREWITT: Once I report into the company they'll take care of me. I'll be all right once I get back.
ALMA: But you'll never make it! There's patrols all over.
PREWITT: I'll make it. I know a shortcut.
Despite being injured in a knife fight and going AWOL, Prewitt's devotion to the army outweighs his personal problems and injuries. Plus, he's concerned for the other soldiers, even if they helped abuse him.
ALMA: What do you want to go back for?
PREWITT: What do I want to go back for? I'm a soldier.
Prewitt's reason might not be incredibly deep, but it gets right to his sense of identity. He can't betray his own identity—and when it's time to be a soldier and fight, he needs to be there.
WARDEN: Sir, this man was a good soldier. He loved the Army more than any soldier I ever knew. I would like to make a formal request that this body be buried in the Army's permanent cemetery at Schofield Barracks.
Here Warden comes full circle. Originally he mistook his duty for just going with the flow and following along with Captain Holmes' unjust treatment of Prewitt. But now he realizes that Prewitt himself was a dutiful soldier, since he was being true to the principles that are supposed to govern military life.