Zim and Professor Membrane have interacted on-screen three times. The first, very briefly, was in the episode "Parent Teacher Night". Although Professor Membrane did talk to Zim, the Irken showed little interest and replied with "yes, yes that's fascinating" before walking off.
Their next interaction would be in "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. While interacting, Zim seems either to be ignoring what Professor Membrane is saying, or understanding what he is feeling as he once replied "Yeah, I know what you mean".
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.
During the events of Invader Zim: Enter the Florpus, Zim kidnapped Professor Membrane and sent him to Moo-Ping 10 while he took over Membrane Labs in his latest attempt to take over the world by manipulating the Membracelets to teleport the Earth into the path of the Irken Armada, so it would conquer the planet.
It is here that it's implied that Zim acknowledges the Professor's potential to benefit the Irken Empire as well as being an enemy, as he told Dib and Gaz that he was the only human capable of reversing his plans and saving the Earth from the Florpus Hole.
The Professor, for his part finally, saw Zim as an enemy but only because he thought that everything that was going on was the result of a hallucination caused by getting hit in the head. Later on, after the fighting was over, he went back to thinking of Zim as his son's little foreign friend, as he believed everything was back to normal.
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".