If you do enclose it, you need to be very careful with safety issues. Monica's woodburner which heats the bricks enclosing it so that enough heat goes out to heat a four-bedroom house shows just how hot the surroundings get.
We knew someone who had, not a woodburning stove, but a big open fireplace with a centuries-old wooden beam as a mantlepiece, in a very old house - It was beautiful, and they used it a lot. One night they went to bed leaving the fire very low and, they thought, safe. One of them was up during the night and when passing the fireplace in a darkened room, realised that the mantlepiece beam was glowing red. It turned out to be almost hollow from the back and would have eventually burned through and collapsed. It could have been a tragedy.