The Doctor's excuse for not knowing whats going on:
The Doctor: "Oh, I don't get out much, bit of a hermit and all that." Bystander: "You missed the gigantic aliens trying to kill us all?" The Doctor: "Is that what that noise was? I was wondering."
The Doctor's reaction upon learning that Misato keeps her sunglasses in a Disney's "Little Mermaid" brand case.
When Asuka accuses The Doctor and Shinji of being in "some sick pedophile thing";
The Doctor (towards Shinji, after denying the relationship): "You're not gonna take that personally are you? Bit of an age difference there, among other issues." Shinji: "That's not actually helping, you know..."
The prequel to Episode 7 opens with the Doctor trying desperately to remember some important problem that he needed to fix. After he finds Shinji attempting to bunk in the TARDIS for the night, he assumes that this was what he needed to handle and promptly tries to sort things out with Misato over a phone call. At the very end, he leaves the console room, only to come rushing back as it turns out the real problem was the TARDIS exploding and taking the planet with it. Which will happen in precisely eighteen seconds.
The Doctor preemptively stops Shinji from giving him the "Passive-Aggressive-Skeppy-McSkeptical" look.
The entire exchange outside the LCL plant after Asuka runs away, from Rei offering to literally drag the other girl back, to Shinji's exaggerated "Fiiiiiiiine", to Shinji running back immediately afterward to question why he suddenly understands German.
After stumbling upon the Rei clones, all of whom are naked, Shinji asks Misato and The Doctor if either of them plan to tell him to look away. When neither of them answer him, he eventually resorts to obscuring his own vision under a fez.
"You're both terrible role models, you know that?"