Though the Mac Pro 4,1 won't run Sierra, the Mac Pro 5,1 will.
Apparently, the Sierra installer only looks for the 5,1 firmware, not the 2009 tag.
The thing about the 4,1 & 5,1 is they are so similar, it's possible to turn one into the other with a simple firmware update.
I have never personally done this, mine was bought already upgraded. The original hack is at http://forum.netkas.org/index.php/topic,852.msg5004.html#msg5004 Which links to a user-made utility to allow this to be changed in both directions, 4,1 to 5,1 & back again. It apparently uses the official Apple Firmware updater as part of the process.
You can pre-empt the download by fetching the 5,1 firmware first, directly from Apple
There are 69 pages of comments... which, honestly, I haven't read. Early theories were that this would allow the machine to use Westmere CPUs - which is now known to be true.
From page 68, this is apparently the process for El Capitan...
- Boot into recovery mode and disable SIP since I'm running El Capitan
- Mount the MacProEFIUpdate dmg (for 5.1)
- Run MacEFIRom's utility.
- Shutdown and power back up by holding the power button until light flashes and beep sounds
- Verify Boot ROM Version: MP51.007F.B03
- Boot into recovery mode and enable SIP
- Reset PRAM
Just to prove it does work, screenshots from my own machine, originally, I think, a Nehalem 4-core, now a Westmere dual 6-core
From Apple KB: About EFI and SMC firmware updates for Intel-based Mac computers
And a comfirmation image from the netkas thread...