score:9
Not much. For the most part the Zone Rouge was never a contested ground.
Let's consider the campaign in four phases and look at each one:
This occurs entirely in the area north of Laon and Verdun and east of St Quentin; so is unaffected by the Zone
The Germans perform this almost without any contact with British and French forces. De Gaulle's armoured counterattack of May 17-19 does cross the Zone, but most fightina again occurs well north of it.
Completely outside (north of) the Zone.
The westernmost (generally running north-south from Lille to Compiegne) section of the Zone is skirted by the German 4th, 6th, and 18th armies, which swing wide to the west of it until between Soissons and Paris.
The Easternmost section of the Zone (running generally west-to-east from Compiegne to Verdun) lies mostly behind the French starting line. The 2nd and 12th Armies, with Guderian's Panzers, assaults across the Aisne well north of the Zone and then diverts around the largest devastated areas to the immediate west of Verdun.
Only in the centre, where the devastated areas are thinnest, do 9th Army, with Kleist's Panzer, group assault across the Zone in the vicinity of Laon and Soissons.
Upvote:6
Adding to Pieter Geerkens' answer, in 1944 the Germans were retreating at high speed when the battle front crossed the Zone, and did not attempt to use it. At that time, doing so would have just caused them pointless casualties, since their units were shattered and disorganised.
Source: Victory in the West, volume II, The Defeat of Germany, the British official history, which doesn't consider the Zone significant enough in context to mention.