Which internal energy source produces heat by converting gravitational potential energy into thermal energy?
This question seems to need a reformulation.
"Which internal energy source [it does make too much sense] produces heat [here we may be talking about stars such as the son and several others that is born and die everyday] by converting gravitational potential [potential energy] energy into thermal energy [here it is okay, energy conservation] ?"
By answering using the best of my knowledge and understanding of the question: pressure.
Pressure is an internal energy source, e.g. in gases here in earth. In stars, the high pressure generated by a high gravitational pull generates heat enough to "melt down" metals, form elements such as hydrogen and He, higher pressures generates even heavier metals and atoms. This is an application of the famous and notorious equation E=mc2.
Example: why Europa, the moon of Jupiter, is not frozen. [go to 28:57]
"He [Subrahmanyan Chandrasekhar] knew that the dense core of a star is under a lot of pressure, and now he began to think about the
fact that pressure is a form of energy." David Bodanis.
Suggested reading : e=mc2 the biography of an equation. David Bodanishttps://www.google.it/?gfe_rd=cr&ei=A_OsVrSGDarD8gfnkp_oAQ&gws_rd=ssl#q=e%3Dmc2+the+biography+of+an+equation