Keep in mind that the cooling devices for computers were designed with having the computer in a case in mind. The airflow over the motherboard chipset and the CPU fan was expected to occur within the case, so cooling "could" become an issue. (Though if your cabinet is open enough I don't see any issues occurring).
This is only true of a small percentage of brand man SFF cases, where the motherboard and heatsink design are laid out with a compact case in mind. Your standard ATX/mATX motherboard is design only with stock heatink fans in mind, where airflow for the CPU fan pushes out across nearby components, the rest of the motherboard is basically left to fend for itself.
Now you average ATX/mATX case has rubbish airflow (including those with 2 dozen fans pointed in random directions) leaving the PC in the case only acts to reduce the effective air volume around the hot parts of the PC, thus the air gets hotter and thus cools the PC less, placing the Case in a cabinet only make things worse as this again acts to reduces the effective volume of air around the heatsink etc.
For example if you have a room temperature of 25 deg C, the heat of the PC arcade monitor, cabinet lighting may bring the temperature in the cabinet up to say 5 Deg above room temperature to 30 Deg C, now the case is now sitting in 30 Deg temperatures so the air inside the case is say another 5 Degs hotter, thus you heatsink fan is now cooling the PC with 35 Deg C air, so your CPU runs at say 50 Degs C. now take the PC out of the Case and put it on a table next to the cabinet in the open air of the room, the room temp may go a up a deg or two after a few hours if all the doors & windows are closed but the CPU is now being cooled with 27 Deg C air so it runs at say 40 Deg C.
Just mounting the PC in the cabinet without a case is going to make it run cooler, vent the Cabinet well top and bottom so hot air can escape at the top and draw cool air in at the bottom, mount the PC as low as possibly away from any heat from the monitor etc, and it will run a lot cooler and for a lot longer than if it's stuffed in a case inside your cabinet. Now of course your PC hardware will effect how much this matters, old HOT running socket-A Athlons & 775 Pentium 4s are going to run a lot hotter than your average Athlon 64 or Core 2 Duo. If your overclocking you might want to duct cool air from outside the cabinet directly onto your heatsink fan.
Note: Heatsinks that use heat pipes often work best when the motherboard is laid flat rather than being vertical.
I am using a case for the jukebox I am building to take advantage of the EMI/RFI shielding that the metal case provides.
Whammoed make a good point on this one the one technical advantage of leaving the PC in the case is that the case acts as a EMI/RFI sheld, protecting both your PC from the outside world and other electronics from your PC. I would be tempted to place a grounded metal shield on the underside of the monitor shelf to protect the PC for any interference from the monitor.