Normally if the sale is conditional on the tenant moving out, then I wouldn't recommend actually closing on the property until that step occurs, however it appears from your post you have already closed on the property so that's not helpful advice at this point. 

Any condition.

The HVAC for the first floor unit supplies conditioned air into the common hallway/stairway.

Another thing is escrow can not technically close until all conditions are met.

This is a condition of the contract and is put into the offer when we make it, usually based on communication with the listing agent.This should be in your contract or in an amendment to the contract. 

The current rents for two of the units are $660 and we want to raise them to $725 and this is fair market rent for the size and condition that the two units are in.

In todays market I think people will overpay for the property in as-is condition as inventory and rates are both so low right now. 

I ask them to ensure the apartment is in tip top showable condition, pay for advertising and pay rent until a suitable tenant that is properly vetted can be found, just as they were.

Are the mechanical system in stable condition?

Generally, material in good condition will not release asbestos fibers and disturbing it may create a health hazard where none existed before.