Did Jonah die and rise from the dead? I've heard from a friend a theory claiming Jonah died in the whale's belly and then he was raised from the dead to preach to the Ninevites.
He told me it is based on those verses:


  "He said: “In my distress I called to the Lord,
      and he answered me. From deep in the realm of the dead I called for help,
      and you listened to my cry." 
  (Jonah 2:2, NIV)
  
  "He answered, “A wicked and adulterous generation asks for a sign! But
  none will be given it except the sign of the prophet Jonah. For as
  Jonah was three days and three nights in the belly of a huge fish, so
  the Son of Man will be three days and three nights in the heart of the
  earth. The men of Nineveh will stand up at the judgment with this
  generation and condemn it; for they repented at the preaching of
  Jonah, and now something greater than Jonah is here." 
  (Matthew 12:39:41, NIV)


I told him I don't agree with his theory because Jonah prayed inside the whale, but he was inflexible.

Also, here is an answer to raise this issue: Jonah and the Whale - Is there any scientific proof that it is possible to survive in the belly of whale?

I would especially like to hear answers from a Pentecostal perspective.
It is possible that Jonah died in the belly of the fish, based on the language used in Jonah 2, but I think most probable that he did not. Notice Jonah 2:5 (2:6 in Hebrew) says "the waters encompassed me up to my neck." That same Hebrew phrase "up to my neck" was used by David in Ps 69:1, “Save me, O God! For the waters have come up to my neck.” The phrase signifies near-death, not actual death. In 2:7, he says Jonah says "my soul was failing me," which again suggests near-death, but not actual death. 

Only in 2:6 do you get language that could be taken to mean that Jonah died. It says "To the bottom of the mountains, I went down.  The land, its bars were behind me forever.  You brought up my life from the pit, O LORD, my God." The first phrase is simply a metaphor for the danger he was in, being dragged to the bottom of the sea. The second phrase is a bit obscure, since the word I have translated "land" is the Hebrew word "erets," which usually means "land, earth," but which some translations have here rendered "underworld." If it were underworld, it could possibly suggest he was momentarily on the other side of the underworld, i.e., dead. However, it is so poetic that, when surrounded by the near-death language on either side, it seems best to take it as a poetic description of the fact that he was on the very verge of death. Besides, I think the usual translation of "land" is more appropriate, since there is no adjective modifying "erets" to suggest it's the "land of the dead" or something like that. It was from this state of near-death that God brought up Jonah's life from the pit.

A former Hebrew professor of mine held the view that Jonah died, so I can appreciate the argument. But I think the argument is based on a misunderstanding of Matthew's comparison of Jonah with Jesus. Jesus only says that the comparison is between the "three days and three nights in the belly of the whale." He does not say the comparison extends to death. I think those who want to say Jonah died are trying to make the two stories of Jonah and Jesus align more than they need to for Jesus' comparison to be valid.