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.”