After Han, Luke, and Leia jump into the trash compactor, why didn't they ask C-3PO to open the door? Even when the walls start closing in they could have just asked him to open the door instead of turning it off.
Why did they ask him to shut down the garbage mashers when they could have asked him to open the door?
Because Luke didn't know the door code and couldn't easily get to the door to find it out.
Brushing distastefully at the clinging slime, he made his way as rapidly as possible toward the hatchcover. Bending, he scraped accumulated detritus away, noting the number thus revealed.
“Open the pressure-maintenance hatch on unit 366-117891.”
Because (in the film) he's reliant on Han reading out the number to him so that he can relay it to C-3PO who can relay it to R2-D2.
Luke saw a hatch against the far wall. To C-3PO, he said, “Hey … hey, open the pressure maintenance hatch on unit number …” He turned to Han and asked, “Where are we?”
Han checked numbers that were etched on the hatch and read aloud, “Three-two-six-three-eight-two-seven.”
C-3PO made sure R2-D2 heard the numbers correctly, and the astromech opened the hatch.
The script indicates that Luke really wasn't in a position to escape even if the door had opened.
INT. DEATH STAR - GARBAGE ROOM:
Meanwhile, Luke is lying on his side, trying to keep his head above the rising ooze. Luke's comlink begins to buzz and he rips it off his belt.
You need to bear in mind that Luke attempts to open the door, and Han tries to blast it, but as soon as they get in, they don't really have time to ask C-3P0 to open the door because there entire episode in the masher is eventful.
I've left out any dialogue to keep it as concise as possible
Han tumbles into a large room filled with garbage and muck. Luke is already stumbling around looking for an exit. He finds a small hatchway and struggles to get it open. It won't budge.
[...]
Han draws his laser pistol and fires at the hatch. The laserbolt ricochets wildly around the small metal room. Everyone dives for cover in the garbage as the bolt explodes almost on top of them. Leia climbs out of the garbage with a rather grim look on her face.
[...]
A loud, horrible, inhuman moan works its way up from the murky depths. Chewbacca lets out a terrified howl and begins to back away. Han and Luke stand fast with their laser pistols drawn.
[...]
Suddenly Luke is yanked under the garbage
[...]
Suddenly the walls of the garbage receptacle shudder and move in a couple of inches. Then everything is deathly quiet... With a rush of bubbles and muck Luke suddenly bobs to the surface.
[...]
Luke seems to be released by the thing.
[...]
Before anyone can say anything the walls begin to rumble and edge toward the Rebels.
[...]
Luke pulls out his comlink.
LUKE
Threepio! Come in Threepio!
Threepio! Where could he be?INT. DEATH STAR - MAIN GANTRY - COMMAND OFFICE
A soft buzzer and the muted voice of Luke calling out for See-Threepio can be heard on Threepio's hand comlink, which is sitting on the deserted computer console. Artoo and Threepio are nowhere in sight.A trooper pushes a button and the supply cabinet door slides open. See-Threepio and Artoo-Detoo are inside. Artoo follows his bronze companion out into the office. ...
The troops hustle off down the hallway, leaving a guard to watch over the command office.[...]
THREEPIO
Use the comlink? Oh, my! I forgot I turned it off!
A New Hope - Script
At this point the trash compactor is almost shut:
The walls are only feet apart... She's terrified and suddenly groans as she feels the first crushing pressure against her body.
So from the script we see that Luke never has time, as mentioned by Valorum, too look at the unit number on the hatch, because of the monster and crushing walls. And when they try to contact 3P0, he's stuck in hiding from the Empire, with the com-link on the table
