I just fixed this same problem with my passenger seat in my 2002 911 C4S. When trying to move the seat up/down I would hear the motor running but the seat wouldn't move. All other forward/backward, tilt, etc seat controls were working just fine. The fix was very easy using the information in this post. Dharn55 had the correct solution--the cable needed to be replaced. You will not be able to duct tape this; the cable needs to be replaced.
Few quick notes:
1. The seat needed to be unscrewed with a E12 socket but does NOT need to come out of the cabin. I just tilted the seat backward after it was unscrewed and used a box to prop it up so I could work underneath it. I had plenty of room to work and since the seat isn't being removed you don't need to worry about tripping any computer faults or power issues as you would if you were to unplug the seat and remove it.
2. I bought a set of external torx sockets at Sears for $9.37 (order online and have it shipped to the store for free, else the Craftsman available in the store if you just "walk in" is expensive: $39.99+tax) -- couldn't find these anywhere else including local auto stores and Home Depot. The two torx screws in the front of the seat have covers which are easily removed by using a screwdriver to lift the tabs and then slide the cover forward (toward the front of the car).
3. The part number listed by Dharn55 996.521.609.00 is correct. I paid $24.97 for it at the local Porsche dealership. Normally I order parts online but this part appears to be special order from Germany so it was actually the same price online.
4. I attached a picture that I found online which I found helpful to understand the controls of the seat. The person that took this picture has obviously taken the seat out of the vehicle (which, again, isn't necessary) but it's a useful picture to orient you to what you're trying to do. The front of the seat (the end closest to the front of the car) is on the floor in this picture. In my specific case, the cable running from the motor on the left was missing (previous owner must have known about this issue but didn't fix it). The right-end of the cable in the picture is attached to the seat gears with a metal clamp (this comes with the part) that you can see if you expand the picture. The left-end of the cable that screws into the motor does not have a metal clamp like this, but rather is held in place between the motor and the metal bracket it attaches to on the seat. The torx screws that hold this in place were much more difficult to get to, especially the one on the left of the motor in this picture....until I got the idea to use the torx bit and a wrench. This was necessary because you have to unscrew from a 90 degree angle because of the way the motor is positioned. You wont be able to unscrew it from the top down, as you normally would, because it is obstructed.
This cable that was replaced is simply a square drive-cable sheathed in white plastic. As soon as I replaced it the up/down functionality of the passenger seat now works again.
Most of the information you would need is already covered in this post but I hope these little extra details help anyone that needs to do this in the future. Thanks 996-99 and other contributors.