Zim and Professor Membrane have only interacted twice on-screen, very briefly in the episode "Tak: The Hideous New Girl". This was when Zim showed up at Dib's house, and the professor told Dib that "his little foreign friend is here" while apparently having made a drink for Zim and was chatting with him. Although Dib told his father several times that Zim was an alien, Membrane only told the two to enjoy themselves, having not listened to Dib's true statement. Though Professor Membrane did talk to Zim in "Parent Teacher Night", Zim showed little interest and replied with "yes, yes that's fascinating" before walking off.
While interacting, Zim seems either to be ignoring what Professor Membrane is saying, or understanding what he is feeling "Yeah, yeah, I know what you mean".
In the cancelled episode "Mopiness of Doom", Zim appeared in his lab disguised as a scientist to make sure that Dib's giving up on defeating him was true. However, Zim and Membrane didn't interact with each other, only listening to the conversation between him and his son. It is unknown what other interactions they would have had, if any at all, in the series finale Invader Dib.