What is risk of further damage to arthritic joints by using pain killers?

None. There is no damage to your joints from taking pain killers. They don't affect your arthritis negatively. Narcotics work on the brain, non-steroidal antiinflammatory medications actually contribute to reducing inflammation, thereby promoting slowing of the arthritic process.
Not Clear. It depends on the pain killer. Old studies have shown that indocin, (indomethacin) an anti-inflammatory could increase joint damage by masking the pain. Films showed progressive joint damage. Using narcotics could in theory cause a charcot like joint - a joint that can be damaged by neurologic loss of protective sensations which we can sometimes see in diabetes and other illness that affect neurology.