Like most people, I hated Dawn in seasons 5 and 6, but by season 7, I liked her well enough. The main problem I had with her after her complaints died down was that she acted so young, and was treated accordingly. It was blatantly obvious that the writers meant for her to be much younger than she wound up being. Notice that, when Buffy was 16 in season 1, she was practically an adult, whereas when Dawn was 16 in season 7, she was still the "baby." I can sort of understand why the Scoobies treated her like a child for so long; they could have been so used to Little Dawnie that they didn't seem to react to her growth; but the writers could at least stop writing her grammatical errors in the script. I can definitely see why Michelle Trachtenberg was getting fed up with her character.
Then the other two that almost everyone has mentioned; Wesley and Cordelia. I hated Wesley on Buffy, but I loved him in every one of his stages on Angel. I adored the Geeky Wes we see in season 1, but I loved the darker Wesley we see in the later seasons even more. Cordelia bugged me at first, but after she started falling for Angel, I started to like her more. I did, however, loathe Evil Cordelia in season 4; at first, it was just my inner Cangel getting increasingly frustrated with the fact that she wouldn't tell Angel how she felt, but after she slept with Connor, I was just thinking, "Augh, what are you doing?!" It made more sense when you found out that she wasn't actually Cordy, but even as I rewatched it, the way she acted annoyed me beyond belief.
I wouldn't say I ever hated Willow, but in season 1, she was easily my least favorite of the Scoobies. I liked her a lot more in season 2, even more in season 3, and, even though she was a little annoying at times when she was grieving over Oz, by season 5, she was my absolute favorite character. I had similar feelings for Angel; I never hated him, but his lack of a personality annoyed me in Buffy. On his own show, I liked him a lot, but he was never my absolute favorite. I have two others; Anya and Riley; but I think this post is already getting a little long.
Since other people have mentioned characters they loved at first but hated later, I'll throw in mine; Buffy, Xander, and (partially) Spike. Buffy was great in high school. She was funny, a little ignorant, and didn't take many things seriously. She slowly started to decline, and during a lot of moments in seasons 6 and 7, I genuinely disliked her. She was so grievy and cocky, and I couldn't stand the way she treated the other characters; especially Spike. I hated seeing her use him. I loved how Xander was the funny man in early Buffy, but I hated the way her reacted when things didn't go his way; and, as more things stopped going his way, he started to annoy me more. I never felt like, "Boo, get him off the screen," but there were some moments where I just got sick of him. Spike was pretty good in season 2, then he was my favorite in season 4; when he fell in love with Buffy, his character slowly started to decline. He annoyed me in Buffy seasons 6 and 7, but luckily, he bounced back and became one of my favorites in Angel season 5.
Wow...I really didn't have to rant for that, did I? Oh well, I did anyway.