Most likely hide glue. Vinegar will soften it....but won't be good for the stain. See if there is some way you can soak the glue off while protecting the rest....maybe with masking tape and saran wrap?

Thanks, lazy gardens. I was recently in touch with someone familiar with this type of furniture, and he told me he believes these pieces were finished with a colored linseed oil. Would you suggest I used a little bit of uncolored boiled linseed oil over the pieces, which might help bring moisture back to the very dry areas? I've also heard good things about tung oil. What would be your opinion on that for these pieces? Thanks.

The home was built in 1989. I have no idea if it was part of the original build. But I think you may be seeing the bottom of the TV that they have mounted above it. I do have another photo showing the entire fireplace that may put its place in perspective. It looks like a woman in front of some sort of plant. I've lived in the South Pacific, and I didnt get that vibe. I thought it reminded me of a middle America, Mayan, or similar region. But I really have no clue. I was thinking that maybe this is a reproduction of a well-known work. I dont even know if this is common on a fireplace or if someone just added it. Thank you for your help.

Agree....but your picture is so dark and shows so few details that I couldn't even give an estimate....has it been painted black? And why do you call ti a priests chair?