It depends on whether the wood is actually singed or just the finish.
You need to remove the finish on the door (and perhaps the whole cabinet unit), but begin with a test stripping of just the damaged area. This can be done with solvent strippers (harsh and smelly, but faster) or citrus based (a bit slower but more person/environment friendly). The wood then needs to be lightly sanded.
Refinish
If the wood is not charred (black or severly discolored), you can strip the rest of the door, sand it, stain it to match the rest of the cabintet and finish it with a polyurethane finish with a sheen similar to the cabinet. Getting it all to match can be hard, so you may wish to do the entire cabinet.
Repair and Refinish
I the wood has a slight char or discolorization, you can try to sand these areas out or use a diluted wood bleach to match the rest of the wood. If this works, then proceed as with refinishing above.
Paint
If the damage is too deep, you need to cover it with an opaque finish, such as paint or a heavily pigmented stain. This will change the overall look so it needs to be done to the entire cabinet. Sand away the damaged finish and lightly sand the rest of the piece with 120 grade sandpaper. Then refinish according to the paint/stain directions.
Countertop
If the laminate is blistered, there is little that can be done except replace it. If it has just lifted, you may be able to reglue it. This generally will not work unless the lifted section is near an edge and can easily be pushed flat. If you can lift the whole section at least 1/2 inch, contact cement works best. Spread glue on both faces to be joined and let dry separately before pressing together. If the gap is too small, you can force conventional glues under the area and weigh it down while the glue sets.
I think I get the picture without one :) I've seen this many times. I'm assuming the wall under the bar you want to cut into is a typical wood framed, wallboard sheathed stub wall. If not, ignore this answer.
You first need to cut out the outer layer of wallboard. If done carefully, no additional finishing will be required, though it may be easier to not be careful and just refinish. It would depend on how easily you think you can match the existing finish. Things will be easier if you plan the opening to fit between two studs, but the opening will possibly be rather small and possibly not in the ideal location. To do things for an ideal size and location, you will likely need to cut out a stud and frame in horizontal 2x4 plates at the top and bottom.
If you're refinishing, remove wallboard from floor to bar top underside and frame the opening properly so the horizontal plates support the new wallboard edges. If you do not want to refinish, place the plates below and above the wall board edges, leaving the wallboard edges unsupported and the plates exposed, they will be mostly covered in the end.
Cut out the wallboard of the opposite surface that is the back wall of the corner cabinet. This time all the edges can be supported. Now you have full access to the depths of that cabinet, but no door. The door can be anything, what I've usually seen is a cabinet face frame applied to the surface of the wallboard, which will cover the exposed horizontal plates. Then a standard cabinet door is attached to the face frame.
If you have decent woodworking skills, you could build the face frame and order the door from a cabinet supply. If not, you could order a plain face frame to match the cabinet door.
It's also possible to case the opening and install a hollow core door, sort of like a little closet. You'd need wide casing to cover the exposed plates. A simple panel door made of MDF could be fabricated and painted. You could just hang some fabric drapes over the opening. The possibilities are only limited by your creativity!
Best Answer
You would (unfortunately) have to sand and stain the whole thing. Stain color is very dependent on a whole bunch of varied factors, and so the final color is always a bit of a guess. In your case, you could reapply stain to the areas you indicated, since they seem to be lighter in shade. Reapplying will darken the color and maybe get it "close enough". Also sand back away from the stained area, but not all the way to bare wood. Just lighten it in a gradient towards the original stained color. Then apply your stain in a similar gradient to get it to blend. That won't perfectly match the color, but it would fade in and hopefully be not-very-noticeable. Exact matching, however, is near impossible unless you sand the whole thing.