Does the heart in Dickinson's "Heart! We will forget him" symbolize warmth and light?

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.

What does "warmth and light" mean in Emily Dickinson's "Heart! We Will Forget Him"?

This poem is a moving statement of a person who is trying to forget about a relationship that has ended or a person trying to move on from unrequited love, or love expressed towards someone who does not return that feeling. As such, the personification of the heart seems to reflect the division of the speaker into the thinking mind and the heart, which is of course the seat of the emotion. With this context taken into consideration, we can understand what is meant when the speaker says:

You may forget the warmth he gave--
I will forget the light!

The two words refer to the positive impact of the relationship that the speaker is now desperately trying to persuade herself to forget. The "warmth" may be refering to the emotional warmth that the man in question brought to the speaker, and the "light" may be referring metaphorically to the way that the relationship helped the speaker to see things in a different way or helped her see and experience things that she had never seen or experienced before. Both of course remind the speaker of his positive qualities, ironically at the same time as she is trying to forget him.

I had to edit your question down to one.

The speaker is trying to forget someone. For whatever reason, the memory of this person is causing the speaker distress. Perhaps this person hurt her emotionally or the speaker just misses this person.

The speaker asks her heart to forget the warmth. The warmth is the emotional connection to this person. If the heart “forgets” this overwhelming feeling, it will be easier for the speaker to put this person out of her mind. The light could refer to the person’s physical appearance which can only be seen in the light. The light could also refer to the mental connection. The memory exists as long as there is light to shine upon it. The memory will fade as the light (on the memory) dims.

Dickinson is using personification by giving the heart human attributes when she asks the heart to consciously forget. When the emotional connections (warmth) fades away, the heart will essentially, but not literally, forget the feeling. This will make it easier for the light that shines on that memory to “dim.”