Because Tak makes her first and only physical appearance in "Tak: The Hideous New Girl", she doesn't have much interaction with Zim, and therefore a proper relationship can't be truly established. However, their feelings towards each other are mainly of a very hostile nature.
From what Tak has revealed about her own personality, she can be described as vindictive and bitter, and capable of holding quite the grudge for a very long time. This is best illustrated by her relentlessness in tracking Zim down to punish him for ruining her chances of becoming an Invader 50 years ago. She even went so far as to follow him to Earth and set up her own base of operations in order to steal his mission, and get the honor she thinks she deserves from the Almighty Tallest.
Zim, although definitely seeing Tak as a nuisance, doesn't seem to take her seriously. Even after she plants explosives in his base and becomes a major threat to his "mission", he fails to see the damage she can really do. When he learns of her plan through MiMi's memory disc, he merely states that "her little joke has gone far enough".
However, he finds that getting rid of her is harder than it seems, and he ends up needing GIR, Dib and Gaz to defeat her. During their final battle outside Earth's atmosphere, Tak attempts to explain that his mission on Earth was a lie, told by the Tallest to get rid of him. In the end, as he makes a report to the Tallest, Zim brushes off Tak's explanation as a "lie."
Early on in "Tak: The Hideous New Girl", Zim doesn't recognize Tak as another Irken, but thinks that she is a normal human girl who has fallen in love with him (a conclusion he draws when he misinterprets her gift of Valentine's meat and a poem as tokens of affection, rather than her making an attempt to blend in and insult him, and cause him great pain to boot, something only Dib takes notice of). Interestingly, Zim chooses to take this opportunity to learn more about the nature of human affection, and attempts repeatedly to win her "affection" - with each attempt leading to suffering on his part. It is debatable whether or not Zim himself has developed a crush on Tak, or whether it's his big ego that leads him to fancy the idea she loves him. However, the latter idea is the more likely.
The writers never establish Zim's feelings towards Tak (Jhonen Vasquez doesn't approve of them getting together out of all the pairings, as he has confirmed countless times), although it is clear that she hates him. Zim also mentions he plans to destroy Tak after using her to study human affection, which implies that he has never cared for her at all.
When Tak followed Zim home, she tried to hypnotize him just as she did with their classmates but he was apparently immune to her hypnotic powers, since he does not react to them; Tak appears confused and angry at this. This opens the possibility that her powers are supposed to work on other Irkens and maybe she even used it with the Tallest. If so, it would explain why they let her prove she can be a good Invader, when they're known to be abusive towards their fellow Irkens.
This could mean that Zim is actually very intelligent (in his own way) and would explain why Tak was angry and confused as she, like the rest of the Irken Empire, views Zim with contempt and believes him to be an idiot. There is also the possibility that Zim cannot be controlled because he is defective, but Tak is equally defective and her powers work on normal humans who have free will.
An early unpublished animation shows Zim kissing her hand out of respect, and her smiling and giving him a big hug before they walk off together, but Jhonen Vasquez immediately disapproved it and it was left out of the final version of the 'love montage'.
Had the series continued, Tak would have reappeared, making her a recurring villain trying to get in Zim's way. In "Top of the Line", she would have entered a competition among the other Invaders, during which their SIR Units would be put through a series of tests (like a dog show or athletic competition). Tak's role would have been to sabotage the other SIRs, leaving all but MiMi, Skoodge, Tenn, and Zim's SIR Units destroyed. Because the script was never recovered, her interaction with Zim during the course of the episode is to be determined.
According to one of the plots for "Invader Dib" (the series finale), Zim would end up punishing Tak to exile on Saturn along with Dib, The Resisty, and the Meekrob. Also, Tak would've been accused of conspiracy to commit genocide on her own people in exchange for Zim's defeat, which he would immediately accuse her of treason. Yet, it's unclear if Tak would have sided with Dib and the Resisty.
After her apparance in "Tak, The Hideous New Girl", her voice appears in "Backseat Drivers From Beyond the Stars", "Dibship Rising", Enter the Florpus and several issues of the comic series through her crashed ship, which has her personality downloaded into it. Zim recognizes her ship during these encounters with it, although he makes no comment on her or how Dib got the ship. In fact, in Issue 46, he seems to not remember who she is when she's mentioned by GIR, despite previously referring to her by name in relation to her ship on several occasions.
Facts of Doom
- Outside of the Invader Zim universe, fans like to ship Zim and Tak together as if they were in a relationship. This pairing is known as ZATR. However, this is non-canon and Jhonen Vasquez has said that he doesn't support it.
- The AI in Tak's ship, being a copy of her mind, gives a pretty good indication of her view on Zim. Namely, that she hates him more than the thought of helping Dib fight him.