I keep my back heel up, and leave my front one down, unless im just bouncing out of range/on the outside or something..
Switch it up adapt to the situation, more often than not tho the back foot is off the canvas, front foot varies. In close i nearly always plant my back foot on the deck. Do what feels most comfortable. Front foot can help with adding a bit of power to the jab if u want too, same with hooks though i planting your left foot when throwing hooks gives as much power.
I'm in the other school... You can't do NOTHING while you're on your heel. You can barely move... you definitely can't punch... so what's the point? And a lot of these guys that move forward on their front heel, also happen to be some of the least defense-minded fighters around. It doesn't matter whether your heel is on the ground or not. You just need to have your weight on the balls of your feet. This is how you can generate the most power more efficiently whether it be walking, running, pushing, jumping, or punching... because you are automatically engaging the calf muscle as well. Having the weight on the heel disengages the calf muscle because you're no longer using the ankle joint effectively.