For cat-backs that have the air tube that feeds the cats, is there any harm to running without the air tube? I have to get the tube welded back up, and don't want to damage the cats in the meantime.
Are you talking about catless intermediate pipes? The cat back doesn't have an air tube on it. Whether or not you do or do not have cats, there is no danger in not running the air tube from the smog pump.
Actually it doesn't make much difference at all. The smog pump just gets the cats up to a temperature a little quicker. As long as you run it through emissions after it's up to temp, it should pass the same. I'll share once experience I had though. A friend of mine had two cats welded into an H pipe that didn't previously have cats on it. The idiots at the exhaust shop only hooked up the air pipe to one of the cats, because it was easier to do that. The cat got so much air it overheated and popped in less than 100 miles.
Kurt is correct, the air pump/smog system is not necessary for the cats. Running without it will NOT ruin your cats.
However, the part where people screw up is when they disconnect the system and don't plug it off properly (ex. weld up the air tube to the H-pipe and plug off the holes at the back of the cylinder heads). If you leave leaks in the exahust system upstream of the cats, raw O2 will get into the exhaust, the cats will over-temp at high enginesloads, and the bricks will melt down. (The beest case when thsi happens is that the cat will just melt into itself and plug up, the worst case is that the molten cat will reaspirate into the head and wreck your engine, but that's more of a concern with modern close-coupled cats and V6's.)