Can there be lasting side effects of self-harm scars?

Yes: For starters, all scars will be permanent. That means you will have an unpleasant reminder on your body when you are a grandmother, long after the pain of whatever it was that caused self-harm has been forgotten. Another danger is that if the problems that cause self-harm aren't addressed, they may escalate. A cut tendon/nerve may not be fully repairable causing permanent numbness or stiffness.

+ psych & cosmetic: Yes, there are very likely to be permanent effects both psychologically and from an aesthetic point of view. Rarely chronic non-healing wounds may undergo malignant transformation but this is more likely in burn scars and called a marjolin's ulcer.