Why do HK chefs let water faucets run unceasingly, when the basin overflows?

This appears to be a dipper well: a continuously running sink used to rinse utensils. The water runs continuously to remove contaminants, always leaving a clean supply of water to rinse dirty utensils and, in this case, to clean out the wok.

They are rather wasteful: the linked article states they use 30-60 gallons (110 to 230 L) of water per hour.

The overflowing water doubles as coolant too, these workburners put out enough BTU to heat the range to unsafe levels.

In a kitchen its all about speed and efficiency after each dish they cook they clean out the wok. Its just easier to let the water run instead of turning it on/off every min.

I want to add that these faucets provide an easy-to-access source of water for cooking. Chinese recipes often call for a small amount of water to be added to the wok during the cooking process. Given the amount of heat put out by those professional Chinese cooking stations, a few seconds to turn on the tap could easily spoil a dish (esp. stir-fries).