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.”