Cut them. off and apply roundup to the stumps. Only need to apply it to the outer growth area (cambian). Will take it into the roots and kill it. I had a big mat of honeysuckle and set it on fire, it never came back.Burn in the spring when it first shows sign of waking up.
Interesting
Burning will kill honeysuckle. But the fire must be hot enough. If you have a clump or a couple of them get a bunch of wood and start a fire on top of them and it'll kill them. But if you have them scattered all over your property its an unpractical way to kill lots of them.
I burn my woods every year. Its a natural fire that is burning leaves, grasses and forbs that have died. Its in the woods so rarely a fire in the woods produces much heat. The fire generally creeps along slowly. These small light creeping fires will kill the tops of small-midsized honeysuckles. But they resprout. The fire will kill very small baby honeysuckles. It won't usually harm a big honeysuckle bush. Of course if conditions are right, really dry, lots of leaves and enough wind to push the fire along so it has more heat it'll kill bigger stuff too. Burning is best done in spring or fall when the woods are dry, low humidity and a decent breeze. As far as killing honeysuckle it doesn't matter.
Burning a woods is good for it. It thins out the woody undergrowth, encourages the growth of forbs and grasses. Damages tick populations and many invasive stuff is harmed by fire while most natives are helped by fire.