No, a PlayStation 2 will not function after its BIOS is dumped. The BIOS is the essential firmware stored on a chip soldered to the motherboard; it handles the boot process, hardware initialization, and basic system operations. Without it, the console cannot start.
Dumping creates a backup copy of the BIOS data (usually for emulation legalities), but this file is not executable on the hardware itself. You cannot "re-flash" a standard PS2 easily, as the chip is often non-rewritable or requires specialized hardware programmers.
Warning: Attempting to physically replace or rewrite the BIOS chip requires advanced soldering skills and will void any remaining warranty.