I can't say I"ve polished out burl before but I've done curly wood and end grain. I don't know that burl would be much different- just more elbow grease working the shellac into the grain.
In Bob Flexner's terms, French Polishing is a build finish - i.e. the beautiful smooth shine comes from multiple layers built-up to a smooth surface and polished to the desired sheen.
In the case of burl, do you really want to fill all the nooks and crannies (just had an english muffin)? Your table top does not look all that rough, but you may still want to consider what effect you'll have if you fill all the spaces. IMHO - it will probably look great!