It is possible. People used to fire pottery in pit fires and things similar to pizza ovens before the days of electric and fuel-burning kilns. It depends on the type of clay and exactly what the oven is made out of.
A low temperature firing or bisque firing would be possible in most pizza ovens. The temperature needed for this process is between 1600 and 1800 degrees Fahrenheit. A fire is capable of reaching these temperatures under normal circumstances without much risk to the oven. For a final firing, however, which requires temperatures of 2200-3000 degrees, could damage the structural integrity of the pizza oven and even start to melt the compounds bricks or burn away the mortar. It depends on what was used to build the oven.
Other drawbacks are that for both types of firings (especially the bisque firing), one must monitor how quickly the temperature rises. One of the purposes of bisque firing, which typically takes about 12 hours, is to finish evaporating any moisture left in the clay before the final firing. Heating it too quickly will cause the moisture to vaporize into steam, which will expand inside the pores of the clay and cause the piece to either crack or explode. For both low temp and high temp firing, cooling to quickly may cause pottery to crack. High temperature firing, particularly if glaze is involved, needs to reach specific temperatures for certain amounts of time in order to harden the pottery and properly melt and adhere the glaze to the piece. This process can take days. If using a wood-fueled pizza oven, this requires 24/7 attention until the firing is complete. If using a gas oven, you must be sure to have enough fuel to last for the duration of the firing.