Well, if you used permanent hair color, you might be able to brighten it up with a red veggie dye, like SFX, but the results aren't going to be nearly as vibrant as if you did it over bleached hair. If the color is somewhat faded, you'll have a better chance.
I'd pick a cooler toned red like SFX Blood Red or SFX Cherry Bomb. I think that either of those would work pretty well over a darker color to bring out red tones. SFX Devilish might work, too, but when I used it, I found that it faded pretty fast in my opinion. If your color is pretty dark, you might want to try to fade it a bit before attempting to "refresh" it with a semi permanent dye. It shouldn't be too hard with a burgundy color. They usually fade really fast.
Check out the section on coloring unbleached hair.
Here are some old pictures of me with partially burgundy colored hair (permanent). It faded quickly and then I went over with SFXBurgundy Wine, but I didn't take pictures. It looked good. Just showing how dark your hair can be and still show good results if you use a dark color.
http://s165.photobucket.com/albums/u77/valkyrie_grrl/ - The first 3 pics