If you have tried to use your head phones and,make sure that they are working in another piece of equipment, you may have to change the head phone jack iPod Classic Headphone Jack & Hold Switch Replacement and here is the part iPod Classic (Thin) Headphone Jack & Hold Switch It is also possible that your headphone jack is stuck in headphone mode...try to plug and unplug your headphones a couple of times since that may free the contacts...
Try the dock connector as well. Place it in a device that plays your iPod, if you have no such thing, may be you can borrow one. If you have no sound from the head phone or the dock connector, check my answer to that particular question and check the link's. http://www.ifixit.com/Answers/View/53334... It might be that the IC is having a bad solder connection. Good Luck and let us know how it is going.
Includes headphone jack & hold switch — 2 available at 14.95 each.