The question here is if you're looking to upgrade to a CPU other than what you currently have.
If you're looking to upgrade to a 2000 series chip (Ryzen 5 2600, for example), then yes, you will need to upgrade the BIOS. But if you plan on keeping your 1600, then you won't need to. Only upgrade the BIOS if you plan to upgrade to Ryzen 2000 series or above.
If you do plan to upgrade to, say, a Ryzen 5 2600, make sure to upgrade your BIOS with your 1600 still installed before you switch the CPUs out.