I would not paint it -- it's a beautiful table. I can't see the extent of the veneer damage from the photo. I would do a burn-in repair on this and it would mostly disappear. That's not an easy method to get equipped and skilled for, though. You might be able to get it filled with a wax filler. Since it's a burl, add a couple of colors and it should blend in fairly well. Level with an credit card or lotto-scratch-off. Then buff with a piece of paper grocery bag. If the veneer is loose, you can drip in a drop or two of CA (super-)glue.first. Here is a link that might be useful: video on how to do it

Paste wax offers minimal protection. While water beads up on it, that is a manifestation of surface tension. It provides nearly zero resistance to water passing through. Polyurethane does not stick well to things. I had to refinish a table like this a few years ago. The owner applied a coat of poly to make it shinier. (File this under: "It sounded like a good idea at the time.") Besides looking terrible, about 1/4 of the table top had poly peeling off. Teak typically has a low-build finish on it. While generally thought to be "oiled" finish, it is usually a low build, very low sheen lacquer. Teak also contains natural oils that inhibit a lot of finishes from curing or curing hard. It needs a little knowledge and extra care in finishing. As I see it, you have a couple of choices if you can't or won't cover it: - Discipline to treat it with more gentleness, use placemats, etc. No finish is immune from physical damage. It's all a matter of degree and type of damage. The strength of the wood and its substrates also becomes a limiting factor. I have seen where people scratch and chip even the "indestructible" finishes. - Take it out and have someone strip and refinish it with a different top coat. What top coat that might be will depend upon the "system" that the place that you go uses, or if you are doing it yourself, what you are able to do based on your skill level, equipment and working environment. Despite this advice, if you choose to apply poly over the top and it does not work, you still have the strip-and-refinish option, it's just that you might be forced into it. The biggest mistake you can make with poly is to apply it too thick and un-thinned. Thin it down up to 50:50 mineral spirits, so you can wipe on the finish with a rag. Very light coats. While you will need more coats to get the same dry thickness, each coat will go on faster and is more foolproof for a beginner than brushing a good finish.

The VOC rules have driven manufacturers away from straight nitrocellulose lacquer. Newer furniture may have any one of a number of 'conversion varnishes' applied to get quick drying and a good appearance like lacquer. Before putting any additional finish on you should try talking to the manufacturer about what finish is already there. There are usually methods that can be used to put additional layers on, but there may be requirements for extra layers to get good adhesion. Even some mechanical roughing (light sanding) may not be enough to get a new layer to stick to factory finishes. Some are very hard and very slick and smooth, with compositions not compatible with commonly available consumer finishes. Vinyl isolation coats can even be needed. Luckily you can now purchase them and do not have to dissolve vinyl pellets in toluene. Manufacturer's information and a real pint store should get you what you need. You can perform some test on the underside with little risk, but should apply the same finish to the top and bottom to minimize warping.

Are you sure it's veneer? It looks like solid wood to me. If so, the end grain would absorb more stain and be darker. In any case you would stain it the same way. If the edges don't take the stain as much, or too much, you could adjust by applying extra coats of stain and/or allowing it to dry longer before wiping.