I don't think he didn't shoot him because he forgave him.
I think the main reason he didn't shoot him was because he has finally escaped Walts control. Throughout the majority of the show Walt had Jesse under his control. Walt manipulated Jesse in many situations in order to benefit himself. An example was when he poisoned Brock to turn Jesse against Gus.
Jesse knew Walt had control over him and manipulated him and it hurt him as I believe Jesse cared for Walt and viewed him as a father figure. This was shown in the epiose "Confessions" where Walt met Jesse and Saul in the dessert when he wants to talk Jesse into leaving town. Jesse breaks down accusing Walt of not caring for him and only wants hime to leave town in order to hinder Hanks investigation instead of benefiting Jesse.
When it comes to the scene where Jesse has the oppurtunity to kill Walt, Walt hands him the gun and asks him to shoot him. I believe Jesse really considerates it as he must hbe harboring hatred toward him. But I think he doesn't shoot him because he wants to break free from Walts control. He does as he leaves without shooting him.