In this poem, the anonymous speaker is trying to get over or move on from a relationship that is characterised by unrequited love. Having loved someone, the speaker now finds that she must, however hard it may be, put that love behind her and move on in life, and forget her love for this man and the way that he made her feel. Thus, to try and help her to do this, she personifies her heart, trying to fashion some division between the will and the emotion and our reasonable, logical minds and our feeling hearts. Consider how the first stanza establishes this:
Heart! We will forget him!
You and I--tonight!
You may forget the warmth he gave--
I will forget the light!
Note how the heart it to forget the emotional warmth that this relationship brought the speaker, and the mind is to forget the way that the relationship changed the speaker's view of the world. Thus the speaker tries to divide herself up into two elements in an attempt to forget the man she can now no longer love.