Planets can be failed stars, but not burnt out ones that we are aware of. Case in point, Jupiter. One school of thought has it that it's a failed star, not attaining enough mass in the early formation of our solar system, to begin the nuclear process. It contains all the right materials to have become a star. That's a gas planet. The rocky ones we know of are here, in our solar system. None have enough mass to start a spark, or have sufficient nuclear material and were never stars. So far, the extasolar planets have been identified as gaseous and a few much larger than Jupiter, possibly large enough to have been a "weak" star, much like a spark (brown dwarf). I suppose, given enough time, and the ability to actually detect one visually, we might see the iron core of a long burnt out star, as this is what many, not all astronomers, think is left after the last bit of energy is used.

No, planets are not just burned out stars. Some may be failed stars, but they are not burned out stars. Some have tried to call Jupiter a failed star, but it really didn't get all that close. It would have had to grown to about 90 times its current size in order to become a star. When a star dies, it becomes one of three types of bodies depending on its mass. Stars remnants that are 0.08 to 1.4 times the mass of the sun (solar masses) ultimately become white dwarves. Stellar remnants of 1.4 to 2.5 solar masses become neutron stars. Those that are bigger than 2.5 solar masses become black holes. White dwarves are made of either helium or carbon. However, the crushing gravity of these compresses the atoms to the point at which it becomes something called degenerate matter. This is a form of matter that is neither solid, liquid, or gas. While the bodies are about the size of Earth, they are not consider planets because they still contain more mass than any planet would. The gravity of neutron stars crushes the stellar remnant until the vast majority of electrons and protons combine to form neutrons. Thus the name neutron star. Neutron stars about the the same size as a moderate sized city. They not a planets. They are essentially gigantic atomic nuclei that are made almost entirely of neutrons. Black holes are stellar remnants that are so massive that their gravities crush then down until they become points of zero volume and infinite density. These are not planets either. So, no, planets are not burned out stars. Planets are bodies that never got massive enough to become stars in the first place.