What did fans think of Wesley?
I'm not sure about the overall fan reaction; I was too young for Star Trek during TNG's original run, and very little of that remains easily accessible on the Internet. Fortunately, Google has archived the old Usenet group alt.ensign.wesley.die.die.die1 which has over 1400 topics (admittedly many, though not most, of which are spambots) and Q-only-knows how many posts, spanning from October 1990 to March 2013. So that's a bit of a clue, anyway.
However, there was definitely enough distaste for the character that Wil Wheaton frequently comments on it. In an interview reported by TrekMovie.com, for example, he said:
What I was hearing back in the old days were older people who were just sort of predisposed to not like a young character on a show. I think the writers could have navigated around that and made him more relatable, instead of like an idea, but they had a hard time overcoming a lot of that stuff. And when I was a kid it was very hard. It was hard not to take that personally. Kids are awkward. Kids are insecure. I spent 50 hours a week doing Star Trek when I was a kid. That was really my life. To go to conventions back then and have people criticizing me and attacking me personally instead of maybe talking about the writing, it was hurtful.
And on his website's FAQ:
Did it bother you that the fans didn't like Wesley?
Yes, at the time, it really really did. Imagine being a teenager, trying to handle all the things a teenager has to deal with. Now multiply that times being on a HUGE TV show, and all these people hate you. It was tough.
However, the reaction wasn't completely negative; Wheaton has commented that some people, particularly those who were kids at the time, watched the show because of Wesley. In a blog post, he writes:
[A]s I got older and started to meet more people who were also kids when Next Generation was in its first run, I started to hear these stories from people, about how they had nothing in common with their parents except for Star Trek, and they wouldn't have watched Star Trek together if Wesley hadn't been on the show. I've lost count of the number of people I've met who are now doctors and engineers and scientists because they were inspired by Wesley and Geordi the way our parents’ generation was inspired by Scotty.
Was Wesley's departure motivated by this dislike?
Unlikely. Wesley wasn't written out of the show per se: Wheaton quit. He tells the story on his website, and it has more to do with how he was treated by the production crew (I've snipped out some of Wheaton's waffling, in the interests of brevity):
After something like this had happened a lot of times, this was finally the last straw: I had been cast by Milos Foreman to be in Valmont. [...] I would have had to sit out the first [TNG} episode of the year, right. That's not a big deal, it's not like I'm the fuckin' Captain, you know. At that point, I was the guy who pushed buttons and said, "Yes, sir!" So, I said to the people on Star Trek, "I need to be written out of this particular episode, because I'm going to do this movie and my film career's going to take off." [...] [T]hey said, "We can't write you out because the first episode of the season is all about you. It focuses entirely on your character and it's your story..." So, he said to me, "The story is entirely about you, we can't write you out." I said, "Well, this really sucks, but I'm under contract to you guys and if that's your call and if that's what you say I have to do, I have to do." I had to pass on the movie.
A couple of days before the season was ready to premiere, they wrote me out of the episode entirely. What they were doing was they were sending me a message. The message was, "We own you. Don't you ever try to do anything without us." That was the last straw for me. I called my agents and said, "They don't own me. It's time for me to leave this show, it's time for me to be gone." That's what really pushed me over the edge. It's not worth it anymore. That's why I left.
1 Nod to Rori in comments for pointing me to this