No:

- A program running inside DOSBox sees the DOSBox BIOS, not the host computer's BIOS

- Even the host computer's BIOS was visible, the host computer's OS is responsible for preventing random programs (including DOSBox) from poking directly at hardware

- I'm pretty sure DOS viruses generally predate flashable BIOS, and even if there was a DOS virus that tried to flash your BIOS, it wouldn't know what to do with a modern UEFI setup

Are you talking about CIH? Anyways it could damage your system on dosbox as dosbox has your files mounted a drive so it could wipe that. Virtual machines are the best way to go. Either way it will not affect the BIOS as most boards are UEFI