I've got a corpse in my fort spreading miasma. I build a casket, placed it, designated it as open to citizens, and almost all dwarfs have the burial skill enabled. It has a description of "This is the resting place of Olon Toraltubul", which is the name on the corpse. Why is the corpse still laying around?
Make sure you have dwarves with the burial labor enabled, otherwise none of them will get the job to move the body to the casket. It's advisable to leave this labor on for all non-critical dwarves.
Check your job list in the manager. Burial is pretty low-priority. If your dwarves have many labors enabled, they may be choosing multiple other jobs before burial, since they pick jobs not in first-come first-served order but by looking at all available jobs and picking the highest priority one. Too many high priority jobs and your smaller tasks will get pushed aside.
I had this same problem, but I had 3 bodies, 5 caskets, 8 idlers, and no names had been applied to any of the graves. My fortress was beginning to have some serious ghost problems, and I was about to create a repeat of this question, but I figured it out so I'll post for other noobs.
May I please emphasis this part of the question: "Designated it as open to citizens."
You have to go to use q to interact with the coffin, and hit b to use it for Burial. You can then use settings for allowing citizens or pets. You can also use the q menu on a coffin to make a room a tomb.