In Emily Dickinson's short poem, "Heart! We will forget him," the speaker uses an apostrophe, speaking directly to her heart. And she also personifies her heart as something that will, with her, forget him. (Hearts don't have minds with which to forget.)
According to Dr. L. Kip Wheeler:
Not to be confused with the punctuation mark, [an] apostrophe is the act of addressing some abstraction or personification that is not physically present: For instance, John Donne commands, "Oh, Death, be not proud."
At the same time, personification is:
A trope in which abstractions, animals, ideas, and inanimate objects are given human character, traits, abilities, or reactions. Personification is particularly common in poetry, but it appears in nearly all types of artful writing. [E.g., The wind howled through the park.]
The speaker talks directly to her heart. Between them—her mind and her heart—they will endeavor to forget "him," perhaps someone who has broken her/their heart. She tells her heart to forget the warmth he gave to her; warmth may be the comfort and pampered feelings she had (at one point) when he was around.
HEART, we will forget him!
You and I, to-night!
You may forget the warmth he gave,
I will forget the light.
The speaker notes that she will work hard to forget the light. This may allude to the sense that in a dark world lacking in love, that his presence, his love provided a beacon, a lighted signal with which to navigate her way.
Emily Dickinson often offers some bit of a surprise in her poems: something playful. Her heart wants to forget his warmth. The speaker is trying, also, perhaps led by the heart, to dismiss the memory of his light from her life.
When you have done, pray tell me,
That I my thoughts may dim;
However, the speaker cautions the heart to tell her when the heart is through so there is no gap in time. Here the speaker implies that she may not be terribly strong: perhaps she is unwilling in trying to forget that light. Here comes the playful twist:
Haste! lest while you're lagging,
I may remember him!
She says that if the heart lags behind too long, the speaker may not be able to stick with her resolve to forget the light, and may actually remember him regardless of their plan. It seems that she may still love "him" and being trying hard to get over him: with no guarantee of success.