Firebrick is not required by code. Here's the language in the IRC code:
R1003.5 Firebox walls. Masonry fireboxes shall be constructed of solid masonry units, hollow masonry units grouted solid, stone or concrete. When a lining of firebrick at least 2 inches (51 mm) thick or other approved lining is provided, the minimum thickness of back and side walls shall each be 8 inches (203 mm) of solid masonry, including the lining. The width of joints between firebricks shall not be greater than 1/4 inch (6 mm). When no lining is provided, the total minimum thickness of back and sidewalls shall be 10 inches (254 mm) of solid masonry. Firebrick shall conform to ASTM C 27 or C 1261 and shall be laid with medium duty refractory mortar conforming to ASTM C 199.
But, there are consequences. The first is you have to make the firebox a little thicker, but that's not a bad idea anyway. We recommend all our Rumfords, especilly the big ones, have firebox walls at least 12" thick. More serious consequences are that ordinary brick and stone are not refractory products and are likely to crack or delaminate when subjected to the thermal shock of being exposed to 1,000 degrees F on one side while the body of the unit is still at room temperature. In the examples above several of the bricks in the red brick firebox failed. Stones in the first stone firebox failed while the soapstone firebox and (reportedly so far) the other stone firebox survived.
If you are a home owner and don't like the look of any firebrick just be prepared for ordinary brick or stone to fail, As our favorite cooking fireplace customer said after five of the brick in the fireback had cracked and the face had spalled off another one, "It's great! The fireplace is only a year old and it already looks as if it's 200 years old."
If you are a mason you have a responsibility to tell your customer that ordinary brick or stone is likely to fail in the firebox - or you will be back every month or so replacing yet another cracked brick.
ASTM C1261 requires firebox brick for residential fireplaces to have a minimum modulus of rupture of 500 psi and a minimum pyrometric cone equivalent (PCE) of 13.
Minerals Zone - fireclay. Information on PCE and classifications of firebrick.
Orton Ceramics Makers of pyrometric cones