It is not a sin to touch a dead body. There were sanitation laws that the Old Testiment had in place that were necessary especially when one did not know the cause of death. (BTW, it is unwise to touch any dead animal or person if you do not know the cause of death and it is advised that if you must that you were very very good gloves and a mask.)
In the Orthodox Faith we do not usually use embalming and have our funerals within 24 hours of the death. We touch the person, we kiss the back of their hands, family and close ones might kiss their cheek or forehead. This is not a "requirement" but it is not forbidden either.
When my grandmother died, I was the one who carefully applied her make up the way she liked to wear it. (Though the funeral home washed her body and hair and dried her hair). My mother arranged her hair the way she liked to wear it instead of the way the funeral home had it. We also put her jewlery on as she always wore three or four necklaces plus a pin and about four rings an a bracelet and watch. (These things were removed before she was buried but for the funeral she was dressed fully in the way she would have wanted to be.)
It was a nice moment when my mother and I did these things for my grandmother. We talked about how much we were going to miss her and how we really wanted her to look "herself" for the funeral. It also meant a lot to the people who came to the funeral as so many of them (I think everyone) said that it was so nice to see her as she normally looked as they had seen her in the hospital or nursing home not really looking like herself.
They kept commenting on how amazing it was that the funeral home was able to get her to look so much like herself.
(My grandmother is not Orthodox, she was embalmed).
But at most Orthodox funerals the person looks natural, they just also look dead, and they look a bit more dead I guess than my grandmother did. But to me, it is the way it should be.
"You ask, will the heterodox be saved... Why do you worry about them? They have a Saviour Who desires the salvation of every human being. He will take care of them. You and I should not be burdened with such a concern. Study yourself and your own sins..." (St. Theophan)
St. Isaac of Syria (7th century): "Someone who is considered among men to be zealous for truth has not yet learnt what truth is really like: once he has truly learnt it, he will cease from zealousness on its behalf."