Yes, forgiving - of course.
Forgetting about it - no way.
You have the responsibility to share your time, emotions and friendship with persons who behave in virtuous ways, and that does not work if you just "forget" about other persons bad habits. So, if things like that happen repeatedly, of course give him the chance to change it by sooner or later confronting him with the fact that such a kind of behaviour is not okay, but ... not to care about it and blindly forgiving things can not be "the way" to handle things like these.
If you'd confront him and he wouldn't care about such hints, it might on the one hand mean that he would be a bad person (generally speaking), and on the other hand that he would be a bad friend (more related to your personal life, but also as a general information towards him), and in that case you should really reflect if someone like this is worth your time and energy. If he would listen and change you would have helped him to improve his habits, and you would both benefit from this
Of course you should not abandon anyone just because s/he did not behave like a perfect human being one single time. But honestly, if someone has a bad character and does not care about that, such a person is not a good company.
To me personally, if I would be misbehaving I would actually prefer to be able to talk about it, for you can not exclude that there was some reason related to YOU that led to his behaviour, and he just didn't dare to tell you about that. So it might be that his lies are, in his opinion, in your interest and a trial not to hurt you.
Anyway, you'll never know such things if you don't dare to talk about it. I guess that's a question on how you want to set your priorities on what is important to you, in your life. To me, truth, if it is truth and not just opinion, is definitely what I am interested in *g*