Just as remembrance is a positive character trait, forgetfulness is a negative trait. There are both practical and spiritual reasons for this.
On a practical level, we might forget debts or obligations that we might have to someone. We might forget to honor commitments that we may have made. And we might forget kindnesses done to us by others, and not show appreciation for such kindness.
On a spiritual level, it is wrong to forget how we have turned away from G-d. We must not minimize sins that we have committed, for otherwise, we will not learn from such mistakes. By conveniently forgetting how we acted wrongly, we may choose to only remember positive things we have done. We will pat ourselves on the back and rationalize that overall, we are pretty much okay.
It is a very natural human tendency to be forgetful. The Orchot Tzaddikim suggests that we first must acknowledge that this is something we must overcome. We should provide for ourselves simple reminders if we owe someone money, for example. One should erect formidable fences so that he not forget the Torah.
Part of our daily routine should be to remind ourselves how fortunate we are to be part of the Jewish nation. We must not forget where it all began on Mount Sinai. We heard G-d speak as we accepted to live in an ethical and moral way, that would be an example for the world.
Forgetfulness can be very harmful. This awareness of its potential dangers, should motivate us to overcome it.