Paris does not die in the Iliad. When the poem ends, with the funeral of Hector, he is still alive in Troy. His death belongs to the rest of the war, told in other ancient poems that were lost and in later summaries.
According to that tradition, Paris was killed by Philoctetes, the Greek hero who had the bow and arrows of Heracles. Philoctetes had been abandoned by the Greeks on the island of Lemnos, on the way to Troy, because of a wound that would not heal. An oracle revealed that the city would fall only with those arrows, and the Greeks went to fetch him. Once healed, he wounded Paris with one of them, and Paris died of the wound. After his death, Helen was given to another son of Priam, Deiphobus.
The irony is that Paris himself had killed Achilles with an arrow. The Iliad does not tell that death, but announces it: Hector, as he dies in Book XXII, prophesies to Achilles the day "when Paris and Phoebus Apollo destroy you, brave as you are, at the Scaean Gates". The archer who kills the greatest of the Greeks also dies by an arrow.