I was hoping someone might have material showing what happened to the remains of the "300", and if the remains of Leonidas ever actually made it back to Sparta, or if they are still there?
What happened to the dead Spartans?
The historian Herodotos tells us plainly what happened to the remains of the fallen at Thermopylai: "they were buried where they fell" (7.228.1). He lists the 3 epigrams that were set up at the site of their tomb, all written by the poet Simonides of Keos. The first celebrated the whole force of 4,000 Peloponnesians who fought at Thermopylai. The second was the famous epigram for the 300 Spartans themselves ("Go and tell the Spartans" etcetera). The third was for the seer Megistias, a guest-friend of Simonides, who had chosen to stay and die by Leonidas' side.
Of course, the simple words "they were buried where they fell" cover a lot of unpleasantness. Since all the Greeks who stayed behind at Thermopylai were killed, there was no one to bury the dead. Herodotos reports that the Persian army piled up 4,000 dead Greeks in one place to show off their achievement to the men of their fleet (8.25.2), but doesn't say whether they did anything else with the bodies. It is quite possible that they left them to rot. Burying or burning that many fresh corpses is a huge task, and the Persians had other matters to attend to, such as the conquest of Greece. It wasn't until the end of the next campaigning season, well over a year after the battle, that the area around Thermopylai was cleared of Persians and the Greeks could finally bury the fallen with proper rites.
By that time there would be little left but bones. Moreover, as I said, the bodies had been piled together by the Persians; Herodotos explicitly says that they dumped Spartiates, allies and helots together in one big heap. As a result, it would have been impossible for anyone to know which of the bones belonged to the 300 Spartans, which to the other Lakedaimonians (perioikoi and helots), and which to the allied Greeks who died in the battle. It's likely that they were all buried together in a single mass grave because there was nothing anyone could do to distinguish them. In any case, it was normal Spartan practice to bury their war dead at the site where they had died, so there was never any effort to pick out their remains and move them back to Sparta. Only their names were commemorated there.
To my knowledge, the mass grave of the Greek dead at Thermopylai hasn't been found. With centuries of intensive use and fortification of the pass, it's possible that there are no traces left.
What happened to the body of Leonidas?
Again, Herodotos tells us what happened in the immediate aftermath of the battle:
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.
-- Hdt. 7.238.1
This would have created a problem for anyone wishing to give the king a proper burial: his bits weren't all in the same place. Like the other Greek dead, his remains would also have been exposed to nature and the elements, making him unrecognisable by the time the Greeks came to bury the fallen. Leonidas isn't separately mentioned in the passage where Herodotos describes the monuments on the mass grave, and there's no indication that anyone at the time knew which bones were his. Indeed, Herodotos says a stone lion was set up as a monument to Leonidas on the hill where he died, not at the place where he was buried (7.225.2).
However, as W.R. Connor put it,^1 "his bones may have been scattered and perhaps permanently lost, but this would not be an impediment to their later "discovery" and restoration". Greek communities told many stories of the miraculous recovery of the bones of mythical and historical heroes, often to explain how the fortunes of a state had suddenly changed for the better. They had little need for proof that the bones in question were real.
It so happened that, about 441-0 BC, the Agiad dynasty faced a PR crisis: their kings had ranged from the underwhelming to the outright treasonous since Leonidas, and it was time for them to reassert the glory of their house. According to the travel writer Pausanias (3.14.1), the Spartan general who had won the battle of Plataia in 479 BC (also named Pausanias) decided to bring the bones of Leonidas back to Sparta so that an annual festival could be held in honour of the hero who embodied Spartan values.
The most obvious problem with this story is that by 440 BC the Pausanias who win the battle of Plataia was dead. The only other Pausanias who may have ordered the recovery of the bones (the grandson of the elder Pausanias) hadn't yet been born. So there's definitely some confusion in the surviving source material about when and why this mission was carried out. But nobody doubts the testimony of Pausanias (the travel writer) that there were two tombs across from the theatre in Sparta in the time when he wrote his guide to Greece: one belonging to Pausanias the victor of Plataia, and one belonging to Leonidas. They may have been set up together around the middle of the 5th century BC.^2
Over the centuries, as the Spartans faded into political irrelevance, they became more and more obsessed with the glory of their past, and stories of the Persian Wars were popular with Roman tourists; as a result, the cult of Leonidas became an ever more prominent part of their festival calendar.
W.R. Connor, 'Pausanias 3.14.1: a sidelight on Spartan history, c. 440 BC?', TAPA 109 (1979), 21-27
P. Low, 'The power of the dead in Classical Sparta: the case of Thermopylae', in Carroll/Rempel/Drinkwater (eds.), Living Through the Dead: Burial and Commemoration in the Classical World (2011), 1-20