Any form of peripheral neuropathy will produce a loss of sensation, including pain. Leprosy, diabetes and nerve injury (axonotmesis) are among the many possible causes. Keep in mind, however, that in the Real World, nerve bundles are responsible for both sensation and movement - that is, both voluntary and involuntary muscle control can be compromised. Muscle weakness or lack of control tend to accompany any loss of sensation (just think of temporary Neurapraxia due to ischemia - when an appendage falls asleep. Not only is it numb, but almost entirely unresponsive as well).
There remains, however, two possibilities for creating a realistic character who cannot feel pain (or much pain, in any case). One is specific damage to the Nociceptors, sensory neurons responsible solely for communicating pain. These are spread throughout the body, so external trauma is unlikely to eliminate them all, but it is not impossible to imagine a viral or bacterial infection targeting and damaging nociceptors system-wide. The subject would still feel pain, due to some other sensory receptors failing to distinguish between stimuli, but it would be very dull compared to the usual response.
The second possibility is damage to the thalamus section of the brain, preventing propagation of the signals to cerebral cortex, or to the cerebral cortex itself. This damage could be the result of an infection or from some past trauma.
edited 28th Oct '10 2:16:04 PM by Keefer