"Fuko has an average Earthling's thought process! Fuko's neighbors often say Fuko is a very normal Earthling!"
The final time we see Fuko is when she is walking with her sister to get a check up at the doctor (she did come out of a ten-year coma: she needs check ups!) and, after some pestering of Kouko, she runs into the forest and approaches a girl under a tree...hence the shot in the opening theme of the first season.
Effect on the Main Plot
I think the most crucial thing Fuko does for the main plot is to draw Nagisa and Tomoya together. Though they had already been working together on the drama club, she brings them together to work on a more immediate goal: her sister's upcoming wedding. An experience of joint effort often brings people closer together: that's why retreats and other events often have teambuilding exercises involving joint effort, such as a relay race or constructing a sculpture. These types of things join people. And that's what carving and distributing the wooden starfish does for Nagisa and Tomoya. Not only is the joint effort a cause of closeness, but Fuko herself works to bring them together. The pre-celebration scene is where this is most shown, I believe. In the scene, Fuko orders Nagisa and Tomoya to call each other by their first names, because they are close enough that they should be doing so already. She succeeds with Tomoya and fails (at least for the time being) with Nagisa, but overall she still succeeds at bringing the two together. And it's an effect they feel after she's gone: in a scene later in the show (I can't at present remember exactly where), Tomoya is calling Nagisa by her first name and then reflects, "When did I start doing this?" It shows that one person's influence can persist, even when that person is lost in memory.
Fuko also leads Tomoya to connect with Yoshino, Kouko's fiancee. Though Tomoya originally meets him in a random encounter, he continues to come into contact with him due to his being engaged and later married to Kouko, who he comes to know because of Fuko. Yoshino is a key player in Tomoya's character development in After Story, so Fuko's influence here is definitely important.
You also first see a light orb (crucial for the ending) in Fuko's arc, when everyone holds the fake class for her, she becomes class leader, and Sanae tells her that all of them are her friends.
In After Story, she acts somewhat as an older sister to Ushio, and, in the process, I think she also helps Tomoya grow closer to his daughter. After all, she sparks a bit of a defense reaction in Tomoya, who tries a few times to protect Ushio from Fuko's strangeness. And there is the final scene, which I will analyze in a separate post (because overall I'm still not completely sure what to make of it).
Now for a couple theories I've heard in regards to Fuko's influence.
One theory is that Fuko is the sheep seen in the invisible world in the beginning of the second episode of After Story. The weight of that theory depends on your interpretation of the invisible world, which I'll cover in a later post.
Another theory, which I think is true, was espoused by Nathan in his recent blog post, and that theory is that Fuko is a foreshadowing of Ushio. He says Kouko and Yoshino are foreshadowings of Nagisa and Tomoya (their appearances seem to indicate this, as the former pair resemble older versions of the latter pair), which I had never thought of before, yet it makes total sense. It's also a good explanation for Yoshino's key role in helping Tomoya take on his husbandly and fatherly duties. In regards to Fuko, her being like a child to Nagisa and Tomoya is obvious. At one point, Nagisa even tells Tomoya, "It almost feels like we're her mother and father." And at the end of the pre-celebration party, Tomoya says that it feels like he, Nagisa, and Fuko really are a family. You can't get more blatantly obvious than that. Thus Fuko does even more for the main plot: she introduces Nagisa and Tomoya to role models that are like older versions of themselves, and she prepares them for the experience of having a child. So all in all, Fuko really works in getting the two prepared to be a family. (Again, I give all credit to Nathan for this insight).
In summary, then, Fuko's effort on the main plot is to bring Nagisa and Tomoya closer together, introduce them to role models, and prepare them for the idea of their being a family.