I work for an automotive dealership as a technician. A few months ago I repaired a vehicle and slightly damaged a part unknowingly that caused a minor oil leak a few months later. The vehicle is in another state now getting repaired. Can my employer charge me for all or part of the repair bill? The vehicle is still under factory warranty also, and that part would be covered under warranty.

As in, deduct from your paycheck? Nope.
http://www.coworkforce.com/lab/WageActFactSheet.pdf

Can they require that I pay for it? Payroll deductions or otherwise.

Some states with such paycheck deduction restrictions also see "forced repayment out of pocket" in the same vein as unauthorized deductions. Give the state DOL a call and see if Colorado is one of those states.

If you don't pay out of pocket, what does the employer say the recourse will be?

It hasn't gone that far yet. They told me they were going to let me know what they decide.

Let us know, OK? Then we can look at your options.