Both are correct, but the first one means the organ, you know, the one that pumps blood trought the body. If you want to say that your heart hurts, but you are talking about your feelings, you must say: Tengo dolor EN EL corazon. Its the exact same, but we use it this way when we are talking about our feelings, like is more poetic/dramatic or something like that.

Both are correctly written but the first one is not expressed in a common way; it could be much better like "siento dolor en el corazón" because "tengo dolor de" is used with more usual kind of pain, like the headache, stomachache or toothache. Anyway, if you want to express that your heart is broken, you could just say "me han roto el corazón". see you!

Me duele el corazón . ;)