I understand in other cultures burying face down is a sign of disgrace and is generally frowned upon. How is it viewed in Judaism? Is the position of the body at burial of concern? If a body is buried face down for whatever reason should it be exhumed and buried face up?
In general body is lying in position with its back on the ground.
Shulchan Aruch : Yore Dea 362:2
נותנים המת על גביו ופניו למעלה כאדם שהוא ישן
We put a deceased body on its back and face is facing towards upper direction, like when person is sleeping
There is a story of a moser (informant) for the Russian government who asked the chevra kadisha that he be buried face down because he wanted to do teshuva. The rav (I forgot who and don't have time to look it up - may have been the Rabbi Moshe Feinstein) stated that they must not do so because of kavod hameis.
After the burial the secret police came and dug up the body to check it. The moser had said that the Jews would bury him face down as a "secret sign" that they were against the government that the moser represented.
Thus we see that it is not proper to have done so.
I found a reference to the story at Ohr Sameach - Ask the Rabbi
Someone had asked about parents who had instructed their daughter to cremate them after death. The Rabbi responded that it was asur and added:
This reminds me of a story, that I find frightening, one which illustrates the frightening evil to which a human can sink and, at the same time, the wisdom of our Sages: Not far from Luban, Russia, there lived a Jew who was a traitor to his people; he spent his life harassing the Jewish community by slandering them to the Russian authorities at every opportunity. At the end of his life, as he lay on his death bed, he summoned the Jewish burial society. "Bury me face down," he said. Then he died.
His most odd request was brought to the great Rabbi Moshe Feinstein. Rabbi Feinstein ruled that, despite the requirement to fulfill a dead person's final wishes, he must nevertheless be buried face-up in accordance with Jewish Law.
A few days after the funeral, the Russian authorities rode into town and summoned the Jewish burial society demanding that the recently buried body be dug up. Mystified, the Jews went out to the cemetery and began digging under the watchful eye of the Russians. When the body was unearthed and became visible, the Russians said, "Enough. Bury him again." Now the Jews couldn't contain themselves: "Please explain why you wanted to dig him up?" they asked.
"This man," the Russians explained, "was our friend. He was forever loyal to us. He told us that the Jews hate him so much for being our friend that they are going to bury him face down!"