The Persians presumably took it all, though we don't know their customs regarding the spoils of war. They may have simply discarded or sold the armour. They left the dead where they were, displaying them to the men of their fleet who were invited to come and see:

Xerxes sent a herald to the fleet. Before sending him, Xerxes had made the following preparations: of all his own soldiers who had fallen at Thermopylai (that is, as many as twenty thousand) he left about a thousand, and the rest he buried in trenches, which he covered with leaves and heaped earth so that the men of the fleet might not see them. When the herald had crossed over to Histiaea, he assembled all the men of the fleet and said: “Men of our allies, King Xerxes permits any one of you who should so desire to leave his place and come to see how he fights against those foolish men who thought they could overcome the king's power.”

After this proclamation, there was nothing so hard to get as a boat, so many were they who wanted to see this. They crossed over and went about viewing the dead. All of them supposed that the fallen Greeks were all Lakedaimonians and Thespians, though helots were also there for them to see. For all that, however, those who crossed over were not deceived by what Xerxes had done with his own dead, for the thing was truly ridiculous; of the Persians a thousand lay dead before their eyes, but the Greeks lay all together assembled in one place, to the number of four thousand.

-- Herodotos 8.24-25

As for Leonidas, his corpse met a particular end:

Xerxes passed over the place where the dead lay and hearing that Leonidas had been king and general of the Lakedaimonians, he gave orders to cut off his head and impale it.

-- Herodotos 7.238.1

The Spartans later went to pick up his remains, carrying them back to Sparta and building a monument to him, where commemorative rituals were performed every year.