In the New York City area (including New Jersey) DO NOT CLOSE UNTIL THE TENANT HAS LEFT!! Both NY and NJ are tenant friendly which means if the tenant has not left before you close, you will have nightmare headaches afterwards and it will be too late then.
Save your sanity and insist you will DELAY the closing until the tenant has moved out PERIOD! This will put the pressure on the seller to make sure the tenant has moved out before closing. Why should you take on the responsibility of possibly doing an eviction on the tenant? I will guarantee that after closing, this tenant will come up with every excuse to stay in this rental and make your life a living nightmare. Let the seller deal with this tenant - not you.
After all, this seller is the one who wants to get rid of this home, so let the seller deal with the tenant! Expect the realtors, lawyers and your lender to be pissed off with you and to start whining, yelling and giving you the finger - all because you refuse to close on time. Tell all these morons - TOUGH and tell them you will be happy to write negative reviews all over the internet. These people need positive reviews and I will guarantee, the minute you mention you will write a negative review - they will shut their mouths. Remember, they are in a hurry to get paid which is why they want you to sign closing documents quickly.