The Man of Steel has thrown down with all manner of enemies, ranging from bald billionaires to fifth-dimensional imps. However, even Superman can find himself punching outside of his weight class, and that just might happen if he ever went toe-to-toe with the all-consuming Phoenix Force.
The powerful cosmic entity first appeared in Uncanny X-Men #101. An immortal being, the Phoenix Force represents the passionate and often consumptive nature of life itself. It is best known for inhabiting Jean Grey during the popular Phoenix Saga, written by Chris Claremont and illustrated by John Byrne. Over the years, the Phoenix Force has taken on a number of hosts, including various X-Men, and most recently, the Avengers.
The Phoenix Force possesses an immense array of powers, including the ability to generate and manipulate intense heat. The full extent of the Phoenix Force, however, is likely beyond the grasp of its mortal hosts. In Secret Wars II, Rachel Summers was able to take on the consciousness of every sentient being of the universe. Jean Grey even used the Phoenix Force to travel through time in order to help Cyclops cope with her loss. However, both feats are relatively rare for the Phoenix Force, who is typically seen as a more conventional - if extremely powerful - being. Against Superman, however, even the Phoenix Force would likely find its match.
The immense heat generated by the Phoenix Force would do little to permanently harm Superman, who draws his power from the yellow sun. Superman does have a weakness to magic, but the Phoenix Force has never quite been understood to be magical in nature, despite its extreme capabilities. Rather, the Phoenix Force is likely more of a representation of cosmic forces, similar to beings like Galactus and the Watcher. As such, the Phoenix Force would draw no such advantage when facing off with the Man of Steel.
This isn't to say the Phoenix Force couldn't destroy Superman, however. While a head-on confrontation would likely result in a stand-still, the Phoenix Force need only turn its attention to the source of Superman's power - the yellow sun. The Phoenix Force wouldn't even have to necessarily destroy the sun either, though such an act wouldn't be out of character for the cosmic being. Instead, the Phoenix would only need to manipulate the sun's energy, switching its output from yellow to red. The transition to a red star would rob Superman of all abilities, leaving him helpless and vulnerable to the flames of the Phoenix Force.
Superman may also have a path to victory in a fight with the Phoenix Force, depending on just who the host is. After all, the Last Son of Krypton is an inspiring beacon of hope, known for his compassionate and forthright nature. Such attributes could motivate the host to resist the Phoenix's control, as Jean Grey did in the climax of The Dark Phoenix storyline. The likelihood of Superman's success would certainly depend on the Phoenix Force's host, as Superman would doubtlessly have more success if the host in question was an ally, like a member of the Justice League, or even someone from his own family, like Jon or Lois. However, these same qualities could motivate the Phoenix Force to strike at Superman all the more, given that he is the last survivor of a doomed world. This could make him an attractive target for the Phoenix Force's world-consuming fury.
As with most hypothetical match-ups, the result depends on a variety of factors, ranging from the exact nature of their conflict to just who is writing the book. The Phoenix Force would be a difficult opponent for Superman to triumph against. In order to win such a battle, Superman would doubtlessly have to trade on any possible connection with the host, even if it meant fanning the flames of one of the most dangerous beings in the cosmos.