This scientific article states that a mutation in the MC1R (which is what causes red hair) can also cause some sort of pain reduction, as well as a lesser sensitivity to anaesthetics.
Results: C57BL/6-Mc1re/e mutant mice and human redheads—both with non-functional MC1Rs—display reduced sensitivity to noxious stimuli and increased analgesic responsiveness to the µ-opioid selective morphine metabolite, M6G. In both species the differential analgesia is likely due to pharmacodynamic factors, as plasma levels of M6G are similar across genotype.
However, this report adds that redheads were more sensitive to thermal pain, while less responsive to the common anaesthetic lidocaine.
Results: Current perception, pain perception, and pain tolerance thresholds were similar in the red-haired and dark-haired women at 2000, 250, and 5 Hz. In contrast, redheads were more sensitive to cold pain perception (22.6°C [15.1, 26.1] vs. 12.6°C [0, 20], P=0.004), cold pain tolerance (6.0°C [0, 9.7] vs. 0.0°C [0.0, 2.0], P=0.001), and heat pain (46.3°C [45.7, 47.5] vs. 47.7°C [46.6, 48.7], P=0.009). Subcutaneous, lidocaine was significantly less effective in redheads, e.g., pain tolerance threshold at 2000 Hz stimulation in redheads was 11.0 mA [8.5, 16.5] vs. >20.0 mA [14.5, >20] in others, P=0.005)
Additionally, this article states that redheads are significantly less sensitive to the anaesthetic desflurane.
The desflurane requirement in redheads (6.2 volume-percent [95% CI, 5.9 - 6.5]) was significantly greater than in dark-haired women (5.2 volume-percent [4.9 – 5.5], P = 0.0004). Nine of 10 redheads were either homozygous or compound heterozygotes for mutations on the melanocortin-1 receptor gene.
Lastly, this Wikipedia article explains...
These observations suggests a role for mammalian MC1R outside the pigment cell, though the exact mechanism through which the protein can modulate pain sensation is not known.
In summary: Redheads do appear to be less responsive to general anaesthetics, as far as reported. However, either an increase and/or decrease to different types of painful stimuli is shown. Also, the mechanism for these results is still unknown, but is very probably related to the mutation in the MC1R which gives redheads their red hair in the first place.
Note: All added emphasis is my own.
Mogil, J, J Ritchie, S Smith, K Strasburg, L Kaplan, M Wallace, R Romberg, et al. “Melanocortin-1 Receptor Gene Variants Affect Pain and Μ-opioid Analgesia in Mice and Humans.” Journal of Medical Genetics 42, no. 7 (July 2005): 583–587.
Liem, Edwin B., Teresa V. Joiner, Kentaro Tsueda, and Daniel I. Sessler. “Increased Sensitivity to Thermal Pain and Reduced Subcutaneous Lidocaine Efficacy in Redheads.” Anesthesiology 102, no. 3 (March 2005): 509–514.
Liem, Edwin B., Chun–Ming Lin, Mohammad–Irfan Suleman, Anthony G. Doufas, Ronald G. Gregg, Jacqueline M. Veauthier, Gary Loyd, and Daniel I. Sessler. “Anesthetic Requirement Is Increased in Redheads.” Anesthesiology 101, no. 2 (August 2004): 279–283.