We can't answer this question in any detail. Herodotos mentions fallen Persians only twice, and in both cases his comments only indirectly tell us what happened with their bodies.

The first one is in the aftermath of the battle of Thermopylai. According to Herodotos, the Persians suffered as many as 20,000 casualties in the battle. Once it was all over, Xerxes invited the sailors of his fleet to come and see the site for themselves - but not before he had dumped 19,000 of the dead in mass graves and hidden them with soil and leaves. The result was that the sailors only saw 1000 dead Persians alongside a much larger number of dead Greeks.

It's possible to take this story at face value and assume that Xerxes was just embarrassed about the way things had gone down at the Hot Gates. But this is a lot of effort just to decieve his sailors, especially when (according to Herodotos) Persian numbers were essentially limitless and a loss of either 1000 or 20,000 wouldn't have made a dent in the size of their army. Why would Xerxes have cared to hide the true death toll - especially since the men who had done the burying could easily have given the game away?

Since Herodotos is more likely to have talked to the (Ionian) sailors from the fleet than to the non-Greeks of the main contingents of Xerxes' army, we should probably assume that the only eyewitness testimony he actually had was that Xerxes showed the sailors 1000 Persian dead. But this was a problem for Herodotos, who worked hard to polish the Spartan propaganda story about an epic 3-day fight in the pass. For his story to have any real credence, the number of Persian dead ought to be much higher. The solution he came up with was to claim that the Ionian sailors only saw 1000 bodies because Xerxes had hidden the rest. In reality we probably have a fairly reliable testimony here that the Persians lost no more than 1000 men in their effort to force the pass at Thermopylai (i.e. fewer than the Greek losses in their futile attempt to hold it).

Either way, the suggestion here is that the Persians would have buried their dead in mass graves. This is the most likely option anyway, since cremating large numbers of bodies on funeral pyres is extremely difficult and requires titanic quantities of firewood. There was no other way to dispose of the bodies in a respectful manner.

The fact that the dead were buried in enemy territory wouldn't have been seen as a problem. At the time of the Persian Wars the Greeks still habitually buried the dead in collective mounds near the site of the battlefield, wherever that happened to be. The Spartans always buried their dead in foreign lands, which they saw as a power move. There's a famous anecdote of an Argive boasting to a Spartan that "There are many Spartans buried in the Argolid", to which the Spartan retorted "There are no Argives buried in Lakonia."

Anyway, the second mention of a dead Persian is after the decisive battle of Plataia. According to Herodotos, something upwards of 250,000 soldiers of the Persian army were killed in this battle. No one really believes this number, but it's pretty likely that thousands were killed, and perhaps tens of thousands, once the Greeks broke into the camp of the defeated Persians and slaughtered the remnants of the army like cattle in a pen. After the battle, there were no Persians left in the area who might bury their own dead. The only surviving elements of the invading army were the ones that had fled the scene. So it was up to the Greeks to decide what to do with the bodies.

Herodotos goes into full curiosity mode here (Histories 9.83.2). He says that the people of Plataia gathered all the bodies in one place, and "once they had been stripped of flesh", various marvels could be seen: a skull of one piece with no sutures; an upper jaw in which all teeth were fused into a single bone; the remains of a man about 2.5m (8ft3) tall. These are the sort of things Herodotos loved to record, since he was interested most of all in the variety and extremes of humankind.

For the purposes of your question, though, what we learn here is as straightforward as it is gruesome. If it was possible for Greeks to observe the bones of the Persians, that means they were not buried; nothing in the story suggests that the flesh was removed by fire. The simplest explanation is that the Plataians dragged all the Persian bodies to one place, piled them together, and left them to rot - "prey for the birds and dogs", as the Iliad says.

The only exception is the Persian commander Mardonios. This great Persian, a cousin of Xerxes, was clearly regarded with a certain degree of reverence even though he was an enemy. Herodotos notes that he fought heroically in the battle of Plataia, leading from the front with his personal guard, and that the Persian line collapsed only when Mardonios fell. The Greek commander Pausanias famously refused to despoil his body like Xerxes had the body of Leonidas. After the battle, his remains became part of their own story:

As for the body of Mardonios, it was removed on the day after the battle; by whom, I cannot with certainty say. I have, however, heard of very many countries that buried Mardonios, and I know of many that were richly rewarded for that act by Mardonios' son Artontes. Which of them it was that stole and buried the body of Mardonios I cannot learn for certain. Some report that it was buried by Dionysophanes, an Ephesian. Such was the manner of Mardonios' burial.

-- Hdt. 9.84

Herodotos' repeated uncertainty about what actually happened shows that several credible sources really wanted to be seen as the people who had buried Mardonios. He mentions only one by name, which may have been the one he trusted most, or the one he heard about first (but it is an open question how a man from Ephesos in Asia Minor could have been the first to lay hands on the body).

Either way, his story also shows that the Persians cared about proper burial, and would have wanted to bury their own dead if they could. Unfortunately, the Greek victory put the bodies beyond their reach. Only the wealthiest Persians would have had the means to track down and reward upstanding Greeks who treated their fallen with respect.