Is the seal a tandem configuration with both seals facing the same direction or a double configuration with the two seals facing opposite directions. For most tandem arrangements, there would still be process gas or buffer gas present to pass between the seal faces. The process gas (potentially dirty) may foul the faces and cause a seal failure. But, it may be reasonable to continue running. For a double seal, there may be no gas flow to pass between the seal faces. This can result in catastrophic seal face failure. We have had this occur on a double dry gas seal on two occasions. We kept running. The rotating seal face fractured and pieces dropped down onto the shaft. When we did shut the machine down, the pieces of Tungsten Carbide had machined through the sleeve and about 1/4" into the shaft. I do not know the hazards of your process. If the seal is leaking to atmosphere and mechanical failure of the seal faces is potentially damaging the shaft, would you want to continue running?
Johnny Pellin