Annie, I'm not Mark....but.
The clue to the "on the grass" method is that the wood has no finish to interfere with moisture absorbtion.
If your piece of veneer has no finish nor any glue on it...it might work.
Remember the concave side is the side that has dried out....the convex side is the more moist side.
Linda c

Annie -- I am Mark -- NO NO NO --on the veneer. If you put veneer out in the yard in the sun -- you might as well try to peel it off by hand, as that will be the result. This method will only work for solid wood PERIOD.....
As far as your problem -- I have had some success with using a old steam Iron with the wood being finish free (again) and steaming the piece and IMMEDIATLY clampimg the loose veneer to the place where it was in the first place.
I think why this works is that the steam softens the old "hide" (dead horses) glue -- so it will stick again.
Good luck!!

So Mark -- you don't have to remove the whole piece, and the curled section is steamed with hot steam like an iron? I don't have to reapply glue? I steam the cleaned wood veneer side and it flattens, heating up the old glue on the back side? Then clamp.
I did take it to a local wood worker and they did tell me to remove the varnish, etc. first and then bring it to them to get it make down -- probably using your method. I'll give it a try. Any suggestions for veneer cracks?
Thanks, Annie

Mark, I have a similar question except the wood cannot be removed ( at least not easily). I took out the backsplash of a 1905 built in hutch to make a pass thru to m y kitchen.
The top of the hutch is made up of 2 oak boards and the one that butts up to the piece of granite that covers the hole where the wall is, is warped at one end..it is concave up and sticks up a little in the middle of the top and goes over 1/8 of an inch below the top of the granite at the extreme edge(most of it is remarkably level with granite edge. I have removed the finish and using hot wet towels got it fairly even. What else should I do? How long should I take to get it smooth and flat? The boards are almost an inch thick and about seven feet long. I can post a pic if you want.

You say "concave up"...concave means that the wood is curved like a bowl...convex means the reverse...humped like the bottom of the bowl.
If the wood at the edge of your pass through is humped up...perhaps it has gotten wet?
Linda C

If you take this table to a wood worker for veneer repair, make sure he uses hyde glue, if the table has any value.

@lindac..yes the top is curved in like a bowl ,but because the middle is sticking up a little and so probably shifted it ends up being lower than the granite. Part of this is maybe the whole thing subtly slopes downward as well. It must have gotten wet..when I am not sure this house is 105 years old..I have had a complete tear off new roof and upgraded insulation done since I ave been here.
@someone..It is a built in sideboard and part of my dining room wall. There is no veneer, solid oak.Let me go take a pic.

It's hard to see but I think you can get the idea

Here is a link that might be useful: