Yes, eye protection is essential when using lasers for welding. Glasses or goggles are required to protect your eyes from both the laser and the welding arc. In addition to eyewear, laser welding safety windows and safety curtains are often necessary to contain the area and ensure overall safety.
As the video illustrates, laser welding can present more than one eye and face protection consideration. The laser radiation itself may be invisible while still hazardous, while the welding process can also produce intense visible light.
It's also crucial that all personnel involved in the laser welding process are trained in laser welding safety and understand the importance of using proper safety equipment.
See here for laser safety welding eyewear options.