You may get less steam than you are thinking, and most of those steamers are not very big. If the steamer is meant to address humidity issues in the house, consider either an electric humidifier for the stove room, or an outside air kit for the stove.
In a more direct answer, the steam won't hurt the popcorn ceiling (because the airflow in that room is significant, because the stove is sucking air out of it). I wouldn't worry about a wooden mantel either as long as it is painted or clearcoated.
I've removed some popcorn ceilings that were impervious to steam altogether. I think one of 'em was stucco.
Next time I have to do that job, I am just taking the drywall down and replacing it.