Don`t know if you got a pic posted or not !! My computer hard drive crashed and working on an old one !! Haven`t been on here in last three weeks !! If its an oil lamp its most likely glass !!They did make pottery lamps but most were fairly new ones !! Look at the bottom, pottery would likely have an unglazed foot where it would sit on a table !! Would help to see it !!
I cannot remember an American lamp in pottery when I was doing shows and buying at flea markets but it has been years since I was an active dealer.
Here is an idea: find a small bulb, e.g., a Christmas tree light bulb, to insert (after taking the burner out) and see if the glow shows through or not.
If it does, it is probably glass.
Is the lamp painted? I cannot help with the burner, Skipinphx :})