Sounds to me like you have a vacuum operated heater shut off valve. This white line is a vacuum line and goes into the control panel that operates the heater. Most earlier Vettes did not have them. What it does is stop hot water from circulating through the heater core when the heat is not on. This reduces the amount of heat in the cabin while you are driving making it easier for the AC to work.
My 73 did not have a valve so I put a manually operated one in myself for the same reason. I have to open the hood to turn it on/off but I only use it in the spring and fall.
Redwingvette2007-11-02 04:16:47______________
Brian - NCM Lifetime Member
73 coupe L48, Flat-top pistons, Performer RPM Heads, Crane Cam and roller rockers, Holley 650 vac sec. Performer intake,
3.55 gear BTO 200-4R trans,
Leather seats, Seatbelt Plus 3point seatbelts, Pioneer CD player
Magnaflow Exhuast System
Dewitt radiator and dual electric fans
Borgeson Steering box