No, you do not have to have everything wired up for the bike to run properly. I just took a look at a CB350F wiring diagram, and it appears that these 4 "warning lights" are all on their own circuits independent of anything that would affect the runability of the engine.
As far as how to terminate the wires... I would personally pull the headlight off and disconnect each one at their respective bullet connectors. Clipping wires and shoving them into holes isn't really my style
.
And on another note. Are you really really sure you want to remove the oil pressure warning light on a bike that's nearly 40 years old? I've saved at least one engine from destruction thanks to that little red light.
EDIT: Nortstudio just replied while I was typing. I also seemed to miss that last caveat in the OP
. I would agree with Nort and 333 that if your bike is not running due to electrical issues, it would be prudent to attempt to return everything the way should be before gutting thing out.