No you cannot use the TurboTax 2015 program to complete and file a tax year 2016 return. Each tax year is a separate program.
You can transfer the 2015 tax data to the 2016 return when you first start a New 2016 tax return using the TurboTax program. See this TurboTax support FAQ on transferring the tax data file using the desktop editions - https://ttlc.intuit.com/questions/1901506-transfer-last-year-s-return-into-turbotax-for-windows
Go to this TurboTax website to purchase a CD or Download of the 2016 desktop editions - https://turbotax.intuit.com/personal-taxes/cd-download/
