Generally it's not an issue. If it doesn't boot, you need to go into your BIOS settings and change the boot sequence to look for the new SATA port first. (Better BIOSes let you choose the boot sequence by device, regardless of which SATA port it's plugged into.) Although the worst BIOSes I've seen won't even auto-detect drives in the ports, and you need to manually enable the new port (and disable the old port).
The only possible issue would be if your boot device is a SSD, and it's an approx 2010-2011 era motherboard. Right around Sandy Bridge was when we transitioned from SATA 2 to SATA 3. A lot of the motherboards from that era had two SATA 3 ports, and four SATA 2 ports. If you're using a SSD you want it plugged into the SATA 3 port. Moving it to a SATA 2 port, while not devastating, does reduce performance. The SATA ports are usually color-coded if you have mixed speed ports. (HDDs are slow enough it won't matter which type of port you plug them into.)