Hi iRFNA,
Before you consider mattress surgery it may be worth reading post #4 here which may be helpful just in case there are other ways that may solve any issues you may be having with your mattress.
As an alternative to disposing of a mattress that is no longer working for you it’s certainly feasible … and can also be very educational and fun, but of course it would be a last resort if there are no other alternatives that would work for you.
It would be helpful to first decide on the overall “surgery” approach or plan (which parts of the pillowtop mattress you want to open up) depending on whether you know what materials or layers that are inside each part of the mattress (the foam above the innersprings or the foam or fibers in the pillowtop itself). Do you have any information about what is inside the pillowtop and what type of padding is used between the springs and the pillowtop?
You can see a little more about a typical pillowtop construction in this video and in this video and this video but there are different ways a pillowtop can be attached to the mattress (staples into foam, hog rings on the springs, or sometimes no internal attachment at all) depending on the construction so it can vary with each mattress. You will find out the specifics as you start “exploring” inside the mattress.
Once you have your “plan” then the first step would be taking apart the parts of the mattress you want to “rebuild” and looking inside to assess which layers may be softening or failing. A pillowtop will generally have a separate “compartment” with foam or fibers in the pillowtop itself and additional foam layers (which may also be lower quality or failing) underneath the pillowtop itself. The layers closest to the top of the mattress (which would be the pillowtop) or any layers that are lower quality in any of the layers above the springs could be the cause of any issues on the mattress.
If you are looking to remove the entire pillowtop then you could use a seam ripper to remove the tape edging between the bottom of the pillowtop and the side panels. You may also need to cut the flange that attaches the pillowtop to the mattress (or remove the staples or hog rings). This would give you access to the layers under the pillowtop.
If you want to only remove the foam layers in the pillowtop itself then you could use the seam ripper to remove the tape edging on the top of the pillowtop itself which would give you access to the materials inside the pillowtop itself and remove the top quilting panel. You may or may not want to re-use the quilting panel on top of any foam you replace inside the pillowtop or the mattress (depending on the thickness of the quilting panel or the materials inside it).
When you have opened up the sections of the mattress you want to explore and assessed and replaced the layers and components that you want to replace, then it’s time to decide on how to finish your rebuilt mattress. If you replaced the foam or fiber inside the pillowtop itself you could use velcro strips to attach the quilting panel on top to give it better shape and attach the side of the pillowtop to the top of the mattress or you could hand sew it back on top. I would make sure that the new design is working well before you do any hand sewing in case you want to experiment with different layers. If you don’t want to reuse the quilting panel (because it’s part of the problem with the mattress) then you could just use your mattress protector and sheets to re-cover the top of the mattress although it won’t have a finished “shape” that looks professional or keeps the pillowtop foam layer inside the pillowtop compartment as well. You could also hand sew a good quality fabric over the new foam to attach it to the sides of the pillowtop to help the pillowtop keep its shape. If you are removing the pillowtop completely and turning it into a “smooth top” mattress you could use the same methods or buy a new zip cover that was the right size and thickness and the type you preferred to recover the mattress.
There are several threads on the forum that discuss mattress surgery including some with pictures posted by westcoaster (here) and by Annmarie (here and here and here in the order of “progression”) and by Nat (here) and by brittany741 here (who did surgery on a Tempurpedic Celebrity) and by litsleeper here (with no pictures but a good outcome).
There are some great instructions for cutting or gluing foam layers in post #3 here and some additional posts with more comments and pictures about cutting foam in post #19 here and post #1 here.
Post #2 here also has some information about removing glued layers.
Some of the better online sources for mattress materials and components I’m aware of are also listed in the component post here.
There are also some mattress surgery threads in another forum here that may also be helpful.
It may also be worth considering having your mattress professionally rebuilt if there is a local manufacturer near you that will do this. This would allow you to finish the mattress more professionally with new tape edging rather than velcro or hand sewing or using a protector or sheets to cover the new layers which won’t fit as tightly around your new mattress and keep it’s shape as well as either a new cover or a professionally rebuilt mattress.
It may take some trial and error to find the layering which works best for you but it can certainly be a worthwhile project for those who are inclined towards DIY projects 
Phoenix