I have an Apple iPhone 4s that is brand new and not activated. Has the warranty period already started for this phone or does it only start after activation?
As far as I know Apple stopped selling the 4s last fall. Therefore the person/reseller you bought it from might have twisted the truth. Where did you buy it? In the U.S.?
You can verify the warranty service and support coverage on: https://selfsolve.apple.com/agreementWarrantyDynamic.do
Apple Warranty starts on the date you buy it (date of purchase). Not on the day you start using it.
WHAT IS COVERED BY THIS WARRANTY?
Apple Inc. of One Infinite Loop, Cupertino, California, U.S.A. 95014 (“Apple”) warrants the Apple-branded iPhone, iPad or iPod hardware product and accessories contained in the original packaging (“Apple Product”) against defects in materials and workmanship when used normally in accordance with Apple's published guidelines for a period of ONE (1) YEAR from the date of original retail purchase by the end-user purchaser ("Warranty Period"). Apple’s published guidelines include but are not limited to information contained in technical specifications, user manuals and service communications.
Source: US-Warranty http://www.apple.com/legal/warranty/products/ios-warranty-document-us.html
If you bought it at Apple and you are not happy with the sale, Apple usually gives 14 days to return the device.
If you do not need it I would suggest returning/selling the device or otherwise start using it. ;-)