In short, that dialogue, more than anything, makes or breaks a fanfic. What do posts like "He would not fucking say that" and "They would NOT have communication skills that good" have in common? Talk. Characters expressing themselves to one another. The faithful recreation of identifiable speech patterns is weighted heavily in the evaluation of a fic's quality. By "speech patterns" I do not just mean the semantic content of a given character's expression, but idiosyncrasies of style and slang, vocabulary and idiom, even gesture, musicality, and rhythm.
Of course believable dialogue is far from the only thing that makes a good fanfic Good. And there are forms of fic writing, particularly highly abbreviated ones like drabbles and ficlets, that in practice tend to de-emphasize its significance. But if we are talking about the romantic, erotic shippy stuff that is the meat and potatoes of online fandom, dialogue does the heaviest lifting short of the consummation itself. Arguably more so! It's the real keystone to the catharsis, and often the catalyst for it. Is there a confession occurring? A provocation? An evasion or ultimatum? Zoom out, big picture: What is the most potent and fundamental mechanic for developing complexity, tension, and transformation within a relationship, getting it to go from one thing to another? Making these two idiots talk to each other! Often clumsily and indirectly and maladaptively, at the worst possible time and in the worst possible situation, about anything or everything but what they should be — but talk they usually do.
What makes fanfic specifically so challenging and rewarding in this regard is that the talking is as much a feat of translation as invention, because both reader and writer are working off an existing model. Liberties taken with plot, form, and even narrative voice have wider buffer zones; you can get creative with circumventing the events of canon while still conforming to its emotional and substantive essence.
But the training wheels come off the moment you open your mouth to speak in another character's voice. And man, nothing will break a reader's immersion quite like he would not fucking say that.