So in effect, the suspect has "lost" or "eluded" the police, and disappeared into an apartment complex.
From a 4th Amendment perspective, that's no different than disappearing into a neighborhood or whole city. Each of the dwelling units is independent as far as the 4th Amendment is concerned; the landlord can't give permission to search the whole facility anymore than the mayor can give permission to search the whole town. They would have to get permission to search per dwelling unit. That's very clearly established in case law.
If the tenants are doing things "by the book", they will not get into even one single unit. See "the book" here in part 3.
So to answer your questions, they can't order tenants out of the unit to search. You don't have to open the door. You can tell them to go away (unless they have a warrant). None of that constitutes "obstructing a search" so you can't be arrested for that merely by denying the search.
You use your words to prevent a police search. If you try to use your body to physically block their movement, then that is obstructing and is a crime.
If a search is illegal, you settle that at the courthouse by getting all evidence drawn from that thrown out. Of course if you have any stuff you're not allowed to have, like Kinder Surprise eggs or an alive person in a cage, you won't be getting that back.
Lastly, there are extreme edge conditions where police might search anyway. If Osama bin Laden was cornered in your apartment complex, or the president's teenage daughter was abducted there, yeah, commanders will decide the damage to their career is worth it, and will search anyway and simply recognize that NOTHING they find will be usable as evidence in court. The FBI has a 10 most wanted list, and they don't do these types of dragnets for any of them. It would be a very, very extreme case.
The other extreme I should cover is, they CAN order you out of a building for a variety of reasons, such as eviction or the building no longer being habitable. That's not a legal reason to search the property OR your bag as you leave. However repairmen, landlord or looters will be tromping through there shortly, and whatever they happen to see incidentally is not protected by the 4th Amendment, and they can report it and you can be busted for it. So appeal to them for 2 minutes to gather some clothing, gather up anything incriminating, throw it in a bag and walk right past them with it.