The past year has been really tough on me, and about 5 months ago, I realized I can't cry. I can feel the emotion, but no tears. When that happens, my heart starts to hurt, as if something was piercing it, and breathing become very difficult and I become dizzy. Is there something wrong with my heart? Or is it something that time will heal?

It could be that you are suffering from hyperventilation, breathing too fast, but it is hard to say from here.
If you have the feeling that a heavy weight is pressing on your chest, or if you get these sensations when you are exercising, walking the stairs or running, then you should consult a doctor right away, it could be a heart problem.