It’s a very interesting question! I think the simplest answer I can give you is no, we don’t need to forgive Satan. I say this for several reasons.
Firstly, God’s great concern in the universe is for people. The context of God’s rdemption and forgivness is the physical world of humanity. That’s why Jesus became a man. He became human, because God wants to redeem humans. Likewise God’s command to forgive one another, just like he forgives us (Matt 6:12) comes within the context of our human world. We are to forgive others because we are just like them - neither better or worse. We are all sinful, helpless, hopeless but cherished and loved by God equally. So we are to treat one another as God has treated us. After all this is the intent of Jesus parable in Mat 18:21-35.
However, Satan, as a member of the spiritual realm, is not a recipient of God’s gift of redemption and forgiveness. He is not like us, he is not human so we are not meant to treat him like a human being. We are not meant to forgive him and God has forgiven us.
Secondly, Jesus attitude to Satan is unforgiving and therefore so should ours. Satan is utterly opposed to everything of God. Rebellion against God (sin) is the defining character of Satan and it is this that Jesus came to destroy (1 John 3:8) - he is not interested in saving Satan! This is the point of the story Jesus told in Mark 3:20-30. He came to bind the “strong man” (Satan) in order to “take his possessions”. He came to fight Satan in order to save us who are captured by him. If Jesus came to fight Satan rather than forgive him then so should we!
Thirdly, Satan has been defeated and his destruction is assured but he is still an enemy looing for victims (1 Peter 5:8). So in response we are told to be wary of him but not afraid of him - to battle against him (Eph 6:12), resist him (James 4:7), wrestle against him and watch out for him (1 Pet 5:8). But never forgive him.
I think when it comes to Satan, we are better off to be respectful of a powerful enemy, but at the same time confident in Christ’s victory over him on our behalf. We shouldn’t ignore him completely, as if he didn’t exist. But neither should we focus too much attention and time on him. He is God’s implacable enemy and therefore he is ours.
Hope that helps.