This is a good question. While Ovid is a humorous poet, this is one of the stories that does not contain too much humor. There are so many other stories that contain humor like the story of Io and Jupiter or Daphne and Apollo.
A summary is a good place to begin. Clymene, Phaeton's mother stated that Helios (the sun god) was his father. Phaeton did not believe and wanted assurance. What could Clymene and Helios do? So, Phaeton finally goes up to see Helios and more importantly gets the right to drive the chariot. Helios is hesitant at first and even says that Zeus could not control the chariot. However, in the end, Phaeton's persistence wins over this father. Helios allows him in the end. The story ends poorly, as one might expect. Phaeton loses control and is on the verge of scorching the world. Zeus has to stop this disaster by hurling a lightning bolt. Phaeton dies as he plunges into the river Eridanos.
In light of this the only possible humor that I can see is the potential use of ironic humor. Phaeton wants truly to live as the son of Helios and winds up dying. Phaeton thinks he can control the chariot, but the chariot controls him. Phaeton wants to make an assent up to the heavens and winds up in the river in death. Finally, Phaeton means "shining." In the end, he is hardly shining.