Firstly, you will probably want to cut the plant off at stump level again. I don't think burning stumps is very effective, or practical. I usually remove the crown, and kill the roots, but there are options.
When I deal with this type of thing, I find that I get great results (but use a lot of energy) using a stump grinder and some full-strength glyphosate. (Note: you shouldn't 'dump' herbicides on stumps. The chemicals are toxic to the environment, and should only be used when called for, and then used as sparingly as possible, only treating where necessary.)
I grind the stump out, dig and remove the chips, drill holes in all the roots larger than 1 1/2 in., and fill with 50% glyphosate. I then (optional) plug the holes with tight-fitting twigs, to minimize soil contamination. Then I put a thick layer of brown corrugated cardboard over the area (just in case) and refill with topsoil.
If you cannot use a stump grinder, and aren't apposed to some physical work, I would dig around the stump, to a level below the crown, and chop it off at the roots (I use an axe). The rest follows as above.
If you don't mind the stump, you can treat with glyphosate and cover with 6-8 mil black contractors plastic. I usually cover that with a mulch, for appearance, but that isn't necessary. If you treat with roundup, the stump will die within a couple months, and don't forget to remove the layer of plastic. You can also leave out the glyphosate, but in that case, the stump may remain alive for more than a season.
Once the stump is dead, you can remove the plastic and cover with organic mulch. This will aid in decomposition (which can take quite a while in salt cedars). Drilling holes in the stump, and applying organic nitrogen also help.
Another option is to spray roundup (not concentrate) on the foliage before cutting, wait until there is complete death, and cut to a stump. The plant, if it regrows, will be very weak. You can put plastic on he stump to stop emergence.