I generally subscribe to "Not broke, don't fix", but there can be legitimate reasons to upgrade. I highly recommend practicing on a test rack before doing it to production equipment.
ControlLogix 1756-L6x processors are being discontinued December 31, 2016. 1756-L7x processors only support down to Firmware level 19, and assuming your program doesn't need the serial port, or a lower firmware version, the L7 is a better option to stock as a spare. Plus Rockwell is up to version 28 already (L6 only supports up to V20).
By comparison Rockwell is only discontinuing the substantially older and more stable PLC-5/80 on July 31, 2017!
There are a million caveats with firmware and Rockwell, and it's much better to do it as a planned upgrade, then on the fly during a breakdown.
Myself I ran into an issue trying to upgrade from V13 to V20 with getting Array subscript to large (Major Fault Type 04 Code 20) which had me very perplexed. Per Rockwell AnswerID 55851 there has been a bug in for-next loops since 2008 with version 16 that persists till at least version 24, and I had to use a workaround SSV (and when I tried to contact Rockwell, they had no interest in telling me if or when it will be corrected.)
Firmware between cards and drives is also a headache which is why I usually disable compatible keying.
Converting from RSLogix 20.01 to 20.03 was also a huge time sink that I lost weeks of my life to for no discernible benefit.
Oh and once you get your processor upgraded to a new firmware you're not out of the woods. If you switch to an L7, your old Panelview Plus HMIs might stop working. They do have Firmware patches to get it working, but Rockwell will scold you for trying to use a PV+ that you bought 5 years ago, and not a PV+6, PV+7, or PV5000 (the later two of which require a new panel cutout).
Rockwell's never-ending upgrade treadmill will make you very tired, but you have to be aware of it, and try to follow along.
With all those Caveats behind you, the basic premise of upgrading firmware should be simple:
-Go online to your processor, save and upload data tables. Save this copy in case you need to revert.
-Download and install the required firmware version. Use ControlFLASH to flash the processor. If you need to revert and flash back down to a really old firmware version, you may need a 32bit OS.
-Open your saved project. In Controller properties Click Change Controller. Change the revision. Eventually it will open up in the new version of RSLogix / Studio. Download to your processor.