Depends on the condition of your walls and the paper. No doubt, the ideal solution is to remove wallpaper whenever possible. Added layers of paper or paint on top of it will just add to any imperfections or problems and cause them to look worse or even for the paper to come off the wall. That's the real concern. If the paper can be removed, it is best to remove it and avoid a bigger pain down the road. Now, that said, I have removed a lot of wallpaper (kitchen as well as every bedroom and bathroom in this house for starters), but my last house had wallpaper that seemed to be superglued to the walls and we could not get it off without destroying the sheetrock. We tried every method known, then sought professional help. They told us to secure or remove any loose edges or corners we might find, float the wall by skimming with joint compound (primarily to smooth out any seam line but also for any torn corners or other imperfections), sand smooth, then paint with the original shellac KILZ. This is one time to not use the water based -- the shellac will go into that paper better and help bind it to the sheetrock. From that point on, you consider it part of the sheerock and move on. You notice I didn't list dining room of this house above? Well, it was fully wallpapered too -- we were just so eager to paint over the the pink paint that we took the outlet covers off and cut those in after the first coat instead of before. The pink paint was covering a pink floral -- it had been like the rest of the house! We sought out the seems and checked them, checked the corners and decided it was secure enough that we were going to leave it. That was 9 years ago. We recently did a wall repair, floated it and repainted -- still looks perfect. Since then, I've hear applying wallpaper and painting over it as a suggestion for creating a smooth, no-texture wall finish and wondered if that would work well if you use a permanent bond to the wall rather than a typical paste that is meant to be removed. Test the wall and the paper. Worst case, new sheetrock is not that expensive (it's the labor to float it that runs up the cost). You don't want anything underneath your new kitchen to cause problems down the road. If it's an open area where you would not have to remove anything to repair, then you are just taking chances with paint and inconvenience. If you have to disconnect, remove, take apart things, then you are talking about more expense, more inconvenience, and the risk of damage or replacement of other parts of your kitchen. Either way you go, proper wall prep is worth the time and hassle -- but that's temporary and the results will be good and can last a lifetime.

If you exposed the seam in the sheetrock, you need to tape it. If you just skim over the seam with joint compound, it will crack at the seam and it will keep cracking no matter how many times you cover it back up (learned the hard way!). I like to use the self adhesive mesh joint tape, it's usually yellow, with setting-type joint compound. It dries/sets fast, and you can speed it up with a hair dryer. Then you can sand it with fine sandpaper, or use a damp rag if you're careful, to smooth it out. The mesh tape can take the sanding a lot better than the paper tape. The paper tape will fray once you hit it with sandpaper, or pucker if you get it wet. You can sand right down to the mesh tape if you want to. This leaves a very thin repair on your wall, and almost certainly won't show once you prime and paint.

I too had a real hard time but I finally found the trick to getting it off easily (no sizing was used on our walls, I am not even sure they were primed!) Anyway, I used paper remover from SW. I put a few ounces in a spray bottle (sort of a super concentrate) and filled with hot water. I used the wall paper tool (wall claw?) (little handle with the wheel that makes little holes in the paper) to go over the section then did it again...the more holes the merrier. Next I sprayed the solution on the area soaking it. Now you walk away for a full 15 minutes. Next repeat spray. Now again, leave for a full 15 minutes. Now spray again and start to peel off paper. It should come right off. You then should spray the area again and wipe off residue. Eventually you will want to clean the walls then prime and paint. Its a lot of work but likely less work than if your painted over paper starts to peel.

Looks like painted over wallpaper to me. I would take a razor knife, cut through the surface and then peel back whatever it is which is coming loose. A little very hot water with a drop or two of dish soap and a few minutes of waiting time will give you an answer on whether it is wallpaper. If it is wallpaper or wall lining paper, removing it is just tedious labor, no special skills needed. If you want to leave it, you will still need to cut the surface and remove whatever is not well adhered to the wall, prime with oil and skim coat over. Drywalling over the surface requires removing all the trim and adding jamb extensions to the windows and doors and reinstalling trim. Not much more money if you were planning to replace trim but an additional expense otherwise. Unless there is significant damage or changes to the existing walls, I wouldn't even consider removing the wall surface; too much labor and money, too little benefit