We're about to start buying small multi-family rentals, preferably those that need some rehab to boost rents.  If the building we buy is already full, how do we get rid of the old renters so that we can rehab the units and then refinance to get our money back?  For example, assume the old renters are paying $300/mo, and we expect to get $600/mo after rehab.

Do we have to officially evict everyone?  Do we have to wait til their current leases expire & then not renew?  Can we raise the rents immediately after purchase, regardless of their old leases, and then wait til they move on their own?  What do the rest of you do?

In most states, leases survive the sale and you would need to honor them until the term expires. 

You cannot evict someone unless they violate the terms of their lease (such as not paying their rent). 

And you cannot raise rents while a current lease is in effect at the lower rate. 

There are some creative solutions, such as what's commonly referred to as "cash for keys" - you negotiate an early lease termination with the tenant, in exchange for some financial consideration/moving money. 

This can work out great and be a win-win if done with some tact. Just be sure to document the agreement in a written document such as a "mutual agreement to terminate lease" (get an attorney to draft it) confirming the date they are vacating/surrendering the unit, and outlining what is to be done with the security deposit and any personal property left behind.

So, barring a "cash for keys" agreement, we have to wait until the lease expires before doing anything.  At that time, though, is it kosher to simply not renew the lease?  Or to double the rent & let them decide whether to leave?

There is nothing wrong in not renewing leases, when they expire.

You need to abide by the terms of the lease for any tenant who is on one that has a timeframe.  If they are month-to-month, you can give them a Notice to Vacate after closing.  Check the landlord-tenant laws for that area as to the number of days and date parameters required.

For either of those options, you could essentially do the same thing at those date frames, but have them sign your lease with doubling the rent.  But, I wouldn't even bother.  I'm sure these are low-income tenants who aren't suddenly going to be able to pull another $300/month out of thin air.  Though they might try to delude themselves and you that they can and that will not end well.  With that said, if you want to make the offers and a tenant takes you up on it, do a very thorough background check.  Gross monthly income that is 3x the new rent.  Check credit reports.  Maybe they took advantage of the low rent they'd been paying to buy a car with a $500/month payment.  Except now they won't be able to pay both.

A lot of people out there live paycheck to paycheck, no matter how little or how much they make.