I made the mistake of mounting my steel frame using the seat tube instead of the seat post. I left three small dents in the frame, nothing that would compromise its integrity. Can steel frames be reforged or repaired to remedy my error? How much would this cost?
It's difficult from your post to figure just how you damaged the frame, or where the damage is.
Possibly you can post a picture?
Small dents or dings in a steel frame should not pose any danger. However, if the damage is to the extent that the frame is creased to the point a crack would form, you might have trouble.
Welding a bike frame can be difficult; the tubes tend to be quite thin.
Brazing is better, but you'd still need to worry about re-finishing and such.
As the other posts said, mere dents in a steel frame aren't usually a problem. A bike shop (preferably one with a gray-haired owner or mechanic who dates back to when steel was common) could tell you for sure.
If you're concerned about cosmetics, ask around for a shop that has "frame blocks." Those are heavy blocks with cutouts to fit around the tubing--you grease the tube, clamp the blocks around the dented part, then rotate the blocks with a wrench to gradually work out the dents (picture a mold that clamshells around the tubing). They'll scratch the paint, but often will get the dent down so it's barely visible. You can find a reference to them here: http://www.bicycleresearchtools.com/aboutus.html
You could probably make your own out of a chunk of oak or other hard wood,
a framebuilder could replace the seat tube - cost would be close to that of a new frame.
remember you'd have to re-paint, too.
if the dents are small enough, you might be able to push them out from the inside, just by jamming a long seatpost or similar in there. not exactly high-tech, and: if there's any kind of crease or a shape that would lead to stress cracks, it's only a matter of time till cracks form. however, it's likely to be a pretty long time.
most likely from the scenario you describe, if they're just cosmetic, put some touch-up paint on and move on having learned your lesson.
(yet another in the very long list of reasons that steel is a better real-world frame material)