Mostly no. Pledges do not commit to any fee but rather they use the "anyone_can_pay" flag so that the recipient can add an extra input to pay the mining fee, or he can use "excess funds" over-the-goal to pay the fee. The page tells him what fee he needs to pay to withdraw his funds, if any, and prompts him to contribute that amount. A fee spike would not change that, he could still just pay whatever feerate is now necessary.

A fee spike would affect automatic refunds though. Those lock in the pledger's fee at whatever feerate was in play when they made their pledge, and if fees are higher than that when the fundraiser ends, their "automatic refund" transactions might not confirm. But pledgers can fix this by returning to the page and manually clicking the Revoke button. That will recreate and replace the refund transaction based on the new fee rates, so no one loses their money as long as they keep their backup file (which contains their private key and everything they need in order to create a new refund transaction).