Yes, American Express will credit you for months that are remaining if you close your Amex Business Platinum credit card account before the membership due date. So for instance if you have 3 months before your yearly membership dues, and you cancel your Platinum card before then, American Express will credit you around $150. If you have other American Express credit card acounts, they can transfer this credit to another account if you like. I think it takes a day or so for the transfer to take place.
Before you cancel your account, just make sure you have a balance of $0.00 and that you do not owe any Membership Rewards points (or you will have to pay for the points). Keep in mind, if you have reward points built up in the account, you may lose them, so if you have a lot of points, you may want to use them up before closing the account (just to be sure, you should ask an Amex rep if you will lose your points).
Canceling an American Express card literally, can take less than 5 minutes (just call the number on the back of your Amex credit card, and speak to a representative). It may be one of the easiest calls you will make, and American Express customer service should not hassle you about canceling the card.
Then within 5-10 minutes of cancelling the card by phone, you can login to your account online and see the card listed as "This account is cancelled".