I need to install a dishwasher with a solid bottom into this space, where this solid pipe (green rectangle above) is in the way. How would I remove the pipe, or at least make it level with the floor, while not damaging the wires inside the pipe?
You have to pull the wire out. Period.
What you're looking at here is the conduit wiring method. It's actually a very excellent way to run wiring, but I gather it's completely alien and unfamiliar to you.
The gist is that you build the pipe route, then you run the wires through it, in that order. The wires can be any color and size you want.
Now, Code requires every junction box cover remain accessible without tools, forever. It can't be covered with drywall, wallpaper, carpet, a screwed down plywood panel, not anything like that. So Wherever the other end of that conduit is, you should be able to access it. That is where you pull the wires back from. Expect them to be individual wires, not the cable you are familar with.
You also see a transition from EMT metal conduit to some sort of flexible conduit system such as AC. I don't know what's going on with the tape, but I assume/hope there is an appropriate coupler there. Also, I don't see what's happening with the ground wire. Perhaps this flexible conduit is not allowed to carry ground (EMT is), and that was their fix. The right solution is to add a ground wire to the inside of that conduit the entire run.
The answer is to reroute that EMT and flexible conduit so it uses a different path. Don't be bashful about getting more EMT or flexible conduit to extend. They sell pre-bent sections and they also have conduit benders.
Do not exit the conduit with bare wires. Honestly I think the metal conduit is a good idea due to the exposure to damage there (from the washer being installed and removed). You are generally allowed to exit conduit with Romex cable, but not here, because it would be subject to damage. However pulling stiff, balky cable through conduit is a major pain, and is often not practical for a DIYer.
So find the other end of this conduit. Bind a rope to your wires at the end. Then pull the wires to pull the rope into the conduit. Then do your work, then use the rope to pull wires back in.
This is a tough one. My first inclination would be to encourage you to re-think your approach either by using some material to elevate the dishwasher or by re-routing the wires if your experience would allow that. Another option could be to remove some of the flooring material using drills or chisels.
However if you feel that cutting the pipe is the best path forward I would first ensure that the wires inside the conduit aren't live before doing anything else. I recommend using one of these as a final check that the wires aren't live. Electricity can kill. I'd encourage you to seek professional assistance if you feel unsafe or unsure.
As for cutting the conduit I would probably use a plumbing style pipe cutter If I was able to get a finger all the way around the conduit. If that proved unworkable I would try a hack saw. Both of these options should give you a very fine degree of control and minimize the chance of damaging the wires.
Once the conduit has been cut I would visually inspect the wires for any damage. If the wires are damaged it could pose a fire hazard. Once that was done I would make sure everything on that circuit was properly grounded. I don't know what you plan to do once the conduit is cut however I would think seriously about if your planned solution is safe and/or up to code.
P.S. This is my first answer on this site and would welcome formatting, style, or any other notes anyone feels I should have.
You can't cut the conduit (pipe) without risking damage to the wires, cutting it will involve sharp things, or high heat, or both.
you could cut the floor and try to bend it out of the way or you could build a false floor above it or find the other end of the conduit and draw the wires back and then cut it.