The request goes away the next day. I think that saying yes and not following through doesn't damage your friendship, it's just that catching that fish increases it. My villagers ask me to catch bugs and fish for them a lot, and I always say yes but never do it unless it fits their house and I'm not planning on adopting them out. I've never noticed a big difference in my friendship levels with them.
- - - Post Merge - - -
I'm not sure about this, but I think that when you make plans to hang out with them and don't show up, that does decrease your friendship level. When you see them again, they usually say something about how they've waited for you but you never showed up.
You can always take notes when they make plans with you (you can use the note-taking app on your 3DS without quitting the game) so you won't forget. If you're planning on quitting the game before that hour, you can set the clock forward and hang out with them before you do that. When I do the latter, I usually only change the clock back the next day, because I've noticed that going back within the same day after visiting or being visited by a villager will sometimes make it seem like it didn't happen at all. Whenever I played the same day over and over to keep villagers in boxes, I noticed that sometimes the same villager would ask me to visit me twice as though they hadn't done that earlier and I wouldn't receive as many letters thanking me for hanging out with them.