I'm going to throw my hat in the ring on this one, and I'll be interested to hear what others think of this opinion. In the areas where your sander turned your wood amber or red, I think what happened is those particular areas stood just a bit proud of the rest of the surface, and you cut through to new wood that had not yet been exposed to the elements. Those spots aren't burns of any sort, they're areas of freshly cut mahogany, exposed by your sander. They look different from everything else on the door because Mahogany changes color to the classic deep reddish brown as it ages. I think the dark strand running down the panel in the first picture is probably the difference between the heartwood and sapwood of the tree this board was cut from, and they aged differently.
If the above is actually the case, I see a few options:
Option one, if you're going to use a finish with no UV protectants, and you're willing to live with the variation for a longish time, the new wood will probably get much closer to the rest of the wood as it ages. This would be the "ignore the problem and it will eventually go away" route.
Option two, if you have the time (or better yet a drum sander of sufficient size, but that would be quite rare) you could take everything down to fresh wood, re-finish with clear coat with UV protectants and move on, the doors would look great. This would be the "swing for the fences" route.
Option three, you could use a gel stain on the doors. Gel stains don't absorb into the wood, they tend to just sit on top so they're pretty darn effective for evening out color variations. The down side is, mahogany is a nicely figured wood and your stain needs to penetrate to highlight its beauty. So this route won't give the very nice wood of the doors it's full impact. Call this the "Good enough I guess" route.
Option four: Stain the doors with whatever stain you choose to highlight the figure of the wood, seal it with a couple coats of dewaxed shellac. Mix some dye into another batch of shellac and use it as a shader to selectively color areas you want to match. After it's dry coat the entire thing with a UV protectant finish such as Marine Polyurethane to help stave off graying and weathering of the wood. Call this the "I'm a chemist" option. This would probably be the one I would attempt myself, but it does assume you have access to spray equipment and are competent using it. Shading (selective coloring of the wood using a colored finish) is nearly impossible to do by brush. The brush or pad lines are too obvious and hard.
OK Cabinetman, you're the guru around here, what do you think?