The answer is: potentially a lot.
Aviation and defense experts say the victims' bodies could contain missile shrapnel. Chemical residue on the plane could confirm the type of weapon that brought down Malaysia Airlines Flight 17. And the location of the wreckage could yield information on how the attack unfolded.
The black boxes could offer vital clues as well. The cockpit voice recorder would record the bang of a missile. The data recorders, which register altitude and position, would be able to tie that information to the timing of a known missile launch in the area.
"You can effectively backtrack and give a relatively high degree of confidence in the location where that missile took off from," said a Manchester, England-based aviation industry consultant, Chris Yates. "If that location happens to be in rebel-held territory, which we all suspect it is, that would be the first point where you could point the finger of blame."
But while anguished families waited to take possession of their loved ones' remains, and investigators had to wait for the rebels to hand over the black boxes, independent observers warned that the pro-Moscow separatists had tampered with the debris and failed to secure the crash site. And the U.S. and its allies fumed that the rebels are trying to cover up evidence they shot down the plane.
Yates warned that the rebels may have already compromised the probe.
"What is gained, of course, is the possibility that whatever evidence remains of a missile strike can be obliterated," he said. "That's the bottom line, I suppose."
In this still mysterious tragedy, for example, the bodies themselves could offer precious clues. A missile from a Russian-made SA-11 mobile launcher, also known as a Buk, would explode outside the target aircraft, hurling shrapnel into the plane. Some bodies might bear the telltale wounds.
"While the stated reasons for removing some of the bodies to a refrigerated train - to protect them from wild animals and slow their decomposition - may be genuine, the bodies, too, are evidence," said Keir Giles, an expert on security at the Chatham House think tank.
Lyubov Kudryavets, a worker at the Torez morgue, told The Associated Press that last Thursday, after the plane went down over eastern Ukraine, a resident brought in the bloody body of a child, about 7 or 8 years old. On Saturday, she said, pro-Russian militiamen came to claim it.
"They began to question me: 'Where are the fragments of rocket? Where are the fragments from the plane?'" Kudryavets said. "But I didn't have any wreckage. ... I swear."
Rebel leader Alexander Borodai has denied he and his comrades-in-arms were trying to tamper with evidence, saying the bodies would be turned over to Malaysian experts.
As of Monday, the remains of 282 people had been reported recovered. A total of 298 people were killed in the downing of the Boeing wide-body jet; some bodies may have been all but obliterated.
A team of international observers suggested that some of the evidence may have been tampered with.
At the biggest site on Monday, "we did not see any perimeter security in the place," Michael Bociurkiw, a spokesman for the Organization for Security and Cooperation in Europe, told reporters in Donetsk. The monitors observed that one of the largest pieces of debris - apparently a large cone section - "had somewhat been split or moved apart."
On an earlier visit to one of the smaller impact sites, where the cockpit and beginning of the first-class section lay, the observers also witnessed apparent tampering.
"We observed workers there hacking into the fuselage with gas-powered equipment," Bociurkiw said.
The alternative explanation for the slow pace of examination and restricted access to the site is simply that a war is going on, said Michael Desch, an expert on international security at the University of Notre Dame.
"I think that what people are missing is that this tragedy has taken place in an active war zone - the Ukrainian Army is today operating against Donetsk - and given that, it is not surprising that the rebels are not being as cooperative as they might otherwise be," he said in an email.
Besides that, eastern Ukraine wasn't known as a model of organization even before the conflict began. The rebel groups that have seized control haven't installed civil institutions that could cordon off the site or organize the orderly removal of bodies.
Russia experts like Andrew Weiss at the Carnegie Endowment for International Peace in Washington said that sheer incompetence, rather than criminal intent, cannot be ruled out as an explanation for the way the rebels are handling the disaster.
"There's just a lot of chaos on the ground," Weiss said. "Everything being messed up is part of daily life. It's not a highly ordered society the way Switzerland is. It's one thing to say it is part of a big conspiracy ... but it's not clear."
Whether by accident or design, the lack of swift access to the crash site may make it harder to determine who and what doomed the jet. And persistent doubts could benefit Russian President Vladimir Putin and undermine the push in the West to impose further sanctions against Moscow.
"It's really a mess," Weiss said. "The question is: Does that mess have some political benefits for Russia?"
Associated Press Writer Lucian Kim in Donetsk, Ukraine, contributed to this story.