I am encountering almost the exact same problem with the socket 1150 i5 5675C Plus GA-Z97M-DS3H motherboard.
CPU fan spins for about 3s, then stops and repeats, but no post.
Tried everything that is usually recommended:
Minimum components: CPU, Memory, PS
Tried the power supply from my other computer.
MB out of the case
Removed CMOS battery
Eventually guessed that the BIOS was not up do date (how can you know before you post?).
Put my i7 4770 chip into the motherboard (requires lower BIOS number) and booted with no problems
Found I was on Bios F5, when the 5675C requires F7
Since there was no way to update the BIOS without windows or DOS (who has a floppy drive anymore?), I installed windows, downloaded the BIOS utility and updated the BIOS to F7.
I Put the i5 chip back in but it still has the same problem.
The Gigabyte website states that the MB is compatible.
If it is not the MB then it suggests the processor is bad, but this very rarely the case.
From what I am reading I am not getting warm fuzzies about the motherboard support for this new chip.
I put a ticket in and we will see...