It depends on the game. Some times I push on through to the end and other times, I'll give up.
List of games I didn't like but finished:
Final Fantasy XIII- I kinda fooled myself into thinking that I liked it but after I finished it, I just felt emptiness due to the embarrassingly poor storytelling and characterization. The definition of style over substance.
Assassin's Creed III- Hated this game. Such a slog. Yet I managed to finish it anyway. In fact, I just started playing ACIV and I'm already enjoying it more than I did ACIII.
Beyond: Two Souls- Inconsistent interactivity with what you can do with Aiden, along with a unfocused story and underdeveloped characters ensures that I felt nothing but apathy for this game.
Star Ocean: Till the End of Time- I actually really liked this game when I was younger but the more I look back, the more I realize that the game really isn't all that great. Unbalanced difficulty, poor characterization, meandering and plodding story, terrible crafting system, etc.
Now here are the games that I didn't like and didn't finish:
Resident Evil 5 and 6- Didn't care much for RE5 and I played RE6 for three hours and found it so boring that I couldn't bring myself to play more of the game.
Demon Souls'/Dark Souls- Tried both of these games but there really weren't for me. Still have a copy of Dark Souls so I might give it another shot if I have the time but I have a ton of other games I rather play.
StarCraft II: Wings of Liberty- RTS games are REALLY not for me and I'll leave it at that.
That's all I can think of for now.