The only reason it is prohibited to wear a “burka” is that it is imitating the goyim (similar to a nun’s habit, which is quite modest, but likewise prohibited). In that respect, wearing a burka is similar to have a tree for the holiday that occurs in late December, or engaging in “trick or treating” for the holiday that occurs in late October, or engaging in celebrations (drunken or somber) in honor of a possibly unhistorical someone the Irish honor in March.
It also might bed prohibited as being something “hadash” in light of a general rule on creating something new.