Will burning kill bush honeysuckle ?

Only if it's small, I tear out the bigger stuff after the rains get rid of the ash. The roots are shallow and then it will die and burn the next time. I pile some in brush piles and burn later too.

I have had great results with buring it. Might take multiple years but it will eventually die from the fires

Burn then spray what comes up. Bush honeysuckle is often thick enough spray won't make contact with all the plants. I burn cattails above the ice for the same reason.