Will they need Cheney to vote?
Cheney cannot vote on cloture, only in the case of a 50-50 tie. Since you need 60 votes for cloture, that won't happen.
Here's where I think Cheney would be needed: cloture fails, the Republicans raise a point of order (Nuclear Option) that only a majority of the 100 is required for cloture on votes to confirm nominations. The parliamentarian sustains the point of order & the Dems appeal that ruling to the full Senate. The Senate votes 50 - 50, The VP in his official capacity as President of the Senate casts the tie-breaking vote & only a majority is required for cloture thereafter (until the rule is changed again.)