What can you remove? Depends on what you want to do. If you want to power the marquee, fan etc. from your PC (usually the same 12V connectors) you could pull out the power supply depending on what sort of transformer you have and if it is directly supplying the 110V to the monitor or not. You could move the connectors for the controls etc. from the control box and just zip tie them to the side of the cab or somewhere else out of the way.
For my Aussie 4 player cabinet (not LAI) I kept the control box but moved it out of the way so I could fit the PC in, I mounted the chassis to the side as high as I could for the same reason and removed the cash box and it's timber platform which took up most of the room. I left everything powered by the cabinet as it was so I can, if I want, still use it to play JAMMA PCB's The only exception to this is the speakers. I wired an amp from a cheap set of PC speakers to the same style of Molex connector as the existing one in the cab so that I can swap the cabinet speakers between the JAMMA connector and the PC soundcard.
A friend of mine with a similar cabinet ripped everything out and completely rewired everything directly to the JAMMA harness to get rid of the box and tidy up the wiring. His cabinet is very clean and tidy but if he ever wants to replace or move something he has to de-solder it all.
I guess I'm just lazy. Here's a few pics so you know what I'm getting at.

The control box was originally in the middle, where it is now was a large shelf under which was the cash box and screwed on top of the shelf was the monitor chassis. By removing that I was able to gain so much more space. I think this is what you need to do in your cab, get rid of the timber shelving/platforms/stands so the inside of the bottom of the cab is clear and then start placing things where you need them to be.

The PC speaker amp looks messy but I couldn't figure out how to mount the PCB without the buttons falling off and where it would be easy to access so just chopped off the top of the speaker box.

Hee you see the J-PAC (player 1 & 2 from the JAMMA harness), Mini-PAC (player 3&4 plus credit buttons for players 1,2,3&4) and to the right the monitor chassis mounted to the inside of the cab. The wiring looks messy because I have paper name tags on both the male and female connectors so It's easy to change player 3&4 back to the regular cabinet wiring with the Molex connectors. (in the first pic you can see the connectors for player 3&4 plugged into the control box with wires hanging out resting on the keyboard).
Good luck with it. If none of this was of any use let me know and I'll remove it to tidy the page up for you.
JJ
edit. Just noticed mattv's post just before mine. I haven't had any issues with power from the cab to the J-PAC but yeah, dont plug it in backwards! The monitors are 110V (US power) hence the need for the transformer.