Use the trim curve function (Edit -> Curve -> Trim...). Select the curve you wish to trim and for the 'bounding object' select the surface then OK the dialog.
Unless you are talking about sketch curves, which may require plan B...
Actually you can trim sketch curves to sheets as well as any others. The only exception is that you'll always generate a copy of the original whether associative or not. What you can't do is affect the contents inside of the sketch with a feature that is outside of it, or vice versa.
For sketch curves you could add an intersection curve with the surface and trim your sketch curves to that. That should work, but I prefer not to make a sketch dependent on an external feature (the dependencies can get messy and come back to bite you later).
