What happens when the DAO doesn't have enough holdings to pay for a passed proposal?

Redemptions that are larger than the DAO's holdings will not go through. You can, however, think of these unfulfilled redemptions as debts that the DAO owes to the redeemer. The redemptions will not expire, and when the DAO does have enough holdings to pay the debt, the redemptions can be successfully executed. 

DAO voters are responsible for not putting their DAO into untenable debt.