As far back as 1916, people recognized the need to protect birds. The United States, Great Britain and Canada and in later years, Mexico, Russia and Japan, signed the agreement which laid the cornerstone of the efforts to protect migratory birds around the world.
There are two sub-species of Burrowing Owl in the United States, the western sub-species and the Florida sub-species. The range of the Burrowing Owl and about 20 sub-species that live in the western part of the Americas from Canada to Terra del Fuego do have owls that migrate. The Florida Burrowing Owls do not migrate, (Who would want to leave the beautiful Florida sunshine???) but they are considered a migratory bird and as such fall under the Migratory Bird Treaty.
The Migratory Bird Treaty Act makes it illegal for anyone to take, possess, import, export, transport, sell, purchase, barter, or offer for sale, purchase, or barter, any migratory bird, or the parts, nests, or eggs of such a bird except under the terms of a valid permit issued pursuant to Federal regulations. The migratory bird species protected by the Act are listed in 50 CFR 10.13.
So before you even think about covering over a burrow that may be on your property, remember that it is against FEDERAL laws and carries severe penalties.