I'm looking at purchasing a three unit property at auction.

One of the three units is occupied by the owner, one is empty and one is rented. What kind of problems can the owner cause for me after I win the property at auction?
How long does it normally take to evict a owner? Can they occupy the empty unit to prolong the process?

You have to give owners a 3 day notice before filing the unlawful detainer (eviction). From there it depends on how backed up your courts are, and whether or not the owner fights it. Figure 60-90 days. There are things owners can do, like damage the property, file bankruptcy or file lawsuits against you and or the lender that can delay things further, but fortunately those things are relatively rare.
I’m not sure with regard to the empty unit. If it is really empty, I see no reason why you couldn’t change the locks and even start renting it, though the prior owner may try to make that difficult, so probably better to plan on waiting. Finally on the unit that is rented, you’ll want to notify that person that you are entitled to receive the rent going forward, and that you are not responsible for their current deposit, if any.
Finally, note that this is based on my experiences, which may not relate exactly to your situation. I would recommend that you check with an attorney before making proceeding with a purchase.

Thank You very much!