Ok, I'm not sure if I can answer if you can run without it or not, I think not, but here's my understanding of the starter solenoid.
You have two big posts on either side, one to the battery, one to the starter. Easy enough. Then you have two small posts, one labeled I the other S. This may be vice versa, but when you turn the key to start the car, it sends juice via the "S" wire (don't recall offhand if thats the brown one or not) to the starter solenoid, the solenoid completes the circuit from the battery to the starter but also completes the circuit to the "I" wire which goes the long way around the engine bay but eventually gets to the coil. What this does is give the coil 12 volts during engine start bypassing the resistance wire which when the car is running normally, drops the voltage to the coil to like 6 or 9 volts, don't recall exactly.
I have heard talk here before that the voltage drop/resistance wire was for the points not the coil, but the basic gist is that if you are running electronic ignition with a high performance coil sometimes it works better if you bypass the resistance wire and run 12 volts to the coil full time. ****I don't know much about this and am only repeating what I think I read a long time ago, so I still have the resistance wire in my car..*****
Regardless of how much voltage you want going to the coil at start or while running, I think you still need the brown wire because I am pretty sure that when you turn the ignition switch to start it disconnects the juice going from the switch to the coil via the resistance wire, and in turn connects the 12V juice going to the coil via the starter solenoid. BUT, I may be wrong on that.
To put it more simply, there are two different circuits in the car providing juice to the coil. One circuit is for starting the car only, the other is for when the car is running normally (any time other than starting). What I wrote here is totally based on my perception that only one of those circuits is providing juice at any given time. If you take away the brown wire you are eliminating one of those circuits and will not be able to start the car.
Assuming you have a stick and can push start it then yes you could "run" the car without that brown wire.
[This message has been edited by sigtauenus (edited 07-12-2001).]