The disposal tube is very similar to a trash tube. It is used to send something to a garbage deposit, or directly eject it to space.
In 2266, during a confrontation between the USS Enterprise and a Romulan Bird-of-Prey, the ship's commander loaded all of his ship's debris into disposal tubes and jettisoned it into space, to fool the Enterprise crew into thinking the Romulan ship had been destroyed. (TOS: "Balance of Terror")
Later in 2266, Captain Kirk dropped a overloading phaser into the Enterprise's disposal tube, which he had found hidden in the red alert light fixture, after declaring a double red alert to evacuate that section of the ship when Spock had heard the distinctive sound of a phaser overloading hidden inside Kirk's quarters. (TOS: "The Conscience of the King")