Air carrier Captain reported that on approach; the First Officer's AP (Autopilot) button became stuck when attempting to disconnect the AP. The button could not be unstuck; which resulted in inoperative pitch trim and continuous AP aural alert during the approach. The First Officer was able to overcome control forces to land the aircraft. Post-flight it was learned that maintenance had applied a placard with adhesive which had caused the AP button to stick.

Narrative

Conducting Captain OE...reviewing logbook I notice MX (Maintenance) had a few issues with the Captain AP (Autopilot)/trim DISC placard. Night prior ZZZ MX had applied a placard; ops check good. I think nothing of it; I'm in the right seat; we move along. Going into ZZZ1; ILS to XX; Captain OE; leg one; properly briefs autopilot coupled approach not approved; so I will disconnect the autopilot at the FAF. Approaching FAF he says gear down; flaps 3; we were doing 180.He clicks off the autopilot; his button is stuck in; which results in the aural 'autopilot; autopilot' to continue. He asks me; can you cancel on your side? I hit my button a couple times and say no I can't cause your button is stuck in. He struggles trying to get his button to release; I tell him; fly the airplane; push the nose over; fly the airplane. He says; I can't get the button out; I say whatever; fly the airplane. We are going to hear this alert the whole way down; while annoying; whatever; block that out and fly the airplane. He pushes the nose over and flies the ILS. I see an AP FAIL caution; but there is no aural alert cause the continuous 'autopilot; autopilot' takes precedence. I call out AP FAIL; disregard and continue; just fly the airplane.This is first leg of Captain OE; so he is left of centerline; but decent touchdown; then the nose slams down. He says I had no trim; I say whatever; get on the brakes. We pull off the runway; with aural alert 'autopilot; autopilot' still going off. We work our way to the ramp through ground and ramp frequency with this alert still yelling at us.We shut down; alert still going off; I reach over to his disconnect button; dig my fingernails into it and get it to come out enough to stop the alert.So; at the FAF we had the autopilot/trim disconnect button stick in. This caused the autopilot alert to continue; but more importantly continued to give a trim cutout signal; which meant our pitch trim was inoperative. The trim was set for flaps 3; gear down; speed 180...it was at 2.5. This meant he had to muscle the elevator to overcome the trim; pulling back excessive forces to maintain glideslope all the way down to touchdown. When I dug the button out I immediately saw the problem. The night prior; ZZZ MX had corrected an AP/Trim DISC placard write up. It appears they applied some homemade placard with a whole ton of adhesive. This adhesive caused the button to stick in (engaged) sending a trim disconnect signal; when it was pressed.ZZZ1 MX dug all the adhesive out with a dental looking pick; applied a proper placard with a couple drops of glue; and we were good to go.My FO (First Officer) (IOE) last week told they had a problem; on his previous trip; disconnecting the autopilot due to excessive glue on a 'homemade' placard.I would like someone to tell the mechanic...here is the problem you caused with your excessive adhesive on some rinky-dink placard you guys manufactured.