For the sake of simplicity, I would like to change "push up" to "open" before I explain. You may change it back to "push up" later.
They heard the door ___________.
C: being opened
D: to be opened
Option A: Though the door cannot perform an action on its own, we will come across sentences such as "I heard the phone ring", "I heard the door creak." and so on. In this case, it is understood that someone called, causing the phone to ring; a wind blew, causing the door to move slightly and creak. The door is a non-living thing and cannot perform an action on its own, but it appears to be as it is responding to a force. Hence, from this perspective, Option A is possible.
Option B: When we use sense verbs (e.g. heard, saw, watched) in the past tense, the subsequent verb is used in the bare infinitive form. Examples: I saw him fall. / I watched him cry. In these examples, "fall" and "cry" are in the bare-infinitive form, meaning without the word "to". Therefore, Option B is impossible.
(Reference URL: http://www.bbc.co.uk/worldservice/learningenglish/youmeus/learnit/learnitv226.shtml
) (Look under the sub-heading: Bare infinitive only)
Option C is certainly possible. By using "being", the sentence is in the passive voice and the focus is on the door being the receiver of an a pushing force. Examples: They heard the door being opened. / They heard the door being pushed up.
Option D uses "to be", which indicates an intention to act in the future. How can they "heard" (past tense) when the action takes place in the future? This eliminates Option D.
I hope this helps you.