I think you could do one of those fire pits on stands if you vented it up through the roof (sort of a vent hood thing that comes down and flares out over the fire area) - sort of like those old 1960-1970 orange round fire pits that people had inside for the "ski" look. Just put in on a couple slabs of brick.
Thats about all I can think of.
Mary

Seems to me the two factors you'd want to consider are proximity to combustible surfaces, including the framing of your porch and the screen material itself; and ventilation, as in not allowing cumbustion gasses to accumulate near the entrance to the house.

My guess is that these issues are probably covered in your local building code, and FP retailers should be able to guide you.

You could certainly have a system meant for indoor use. You may find there are cheaper and simpler solutions as well.

My guess is that there is not anything made for "outdoors" since this screen porch is actually "indoors".

Lennox offers a beautiful stainless steel model fireplace that is specifically designed for outdoor use. It is a zero clearance unit that you would have to build framework for out of standard 2x4 construction materials and then face it with whatever material you desired such as cultured stone brick or natural rock. I know that if you take an "indoor" appliance such as a stove and place it in a screened back porch setting the manufactuer will more than likely void the warranty I had a customer that wanted to do this and vermont castings said no way. But there are options out there. Just have to find them. Hope this helps you out a bit