Yes, and no. Open burning in the City of Austin requires a permit from the Fire Marshal's Office. Any Open Burning without a permit is prohibited, and may result in a citation. This includes the burning of trash, rubbish, yard clippings, tree trimmings, etc.
Open Burning is defined in the Fire Code as:
“… The burning of materials wherein products of combustion are emitted directly into the ambient air without passing through a stack or chimney from an enclosed chamber. Open burning does not include road flares, smudgepots and similar devices associated with safety or occupational uses typically considered open flames or recreational fires. For the purpose of this definition, a chamber shall be regarded as enclosed when, during the time combustion occurs, only apertures, ducts, stacks, flues or chimneys necessary to provide combustion air and permit the escape of exhaust gas are open.”
Recreational Fires are defined in the Fire Code as:
“An outdoor fire burning materials other than rubbish where the fuel being burned is not contained in an incinerator, outdoor fireplace, barbeque grill or barbeque pit and has a total fuel area of 3 feet (914 mm) or less in diameter and 2 feet (610 mm) or less in height for pleasure, religious, ceremonial, cooking, warmth or similar purposes.”
Requirements for Recreational Ground Fires and Portable Outdoor Fireplaces.