Because the Sox and Bulls have different boards of directors I'm very sure JR can't (or won't even if he could) take money from the Bulls to give to the Sox.
As far as kicking in his personal fortune? Again I'm very confident the answer to that is a resounding no. JR is a shrewd businessman first, second..always. He'll never be confused say with Artie Moreno or Mike Ilitch as far as being willing to use his personal money in the quest to win. That's not him.