I hear this a lot in song lyrics, and I just can't figure out what a "hot chest" is supposed to mean.
As for the definition 胸, it means chest is partially correct. To tell you the truth, 胸 has various meanings besides chest.
In my dictionary, 胸 is defined as:
1. the chest, the breast, the bosom
2. the breasts
3. the heart, the lungs
4. one's heart, one's mind
In the case of 熱い胸, 胸 means neither some organ nor part of the body, but the place where man's feeling is supposed to exist.
Literally the phrase means the hot heart, and it means elevated emotions.
In the site shown by binom's comment, there is a sentence "胸が熱くなる", that is a sentence closely corresponding to "熱い胸", which means "someone is moved by something so they become in the state of elevated emotions."