To burn something one needs: fuel, an oxidizer, and a high temperature.
The atmospheres of the gas giants contain *no* free oxygen - and have 'fuels' at relatively low concentrations. So a hot plume of a rocket exhaust cannot, in any way, set light to Jupiter.
(for example)
But more crucially, stars do not 'burn' chemically - they require vast pressures and temperatures at their cores to initiate nuclear reactions. These pressures and temperatures scale entirely with the mass of the object: the gas giants that we have in our solar system are barely a few % of the mass needed to create conditions that support fusion.
The minimum mass to create a star is .08 solar masses. This allows the pressure at the core to begin fusion of hydrogen into helium. Let's say there are other gasses, other than hydrogen; flammable gases. It would depend upon the overall mass and the nature of the gas as to when it would become a star,
But more crucially, stars do not 'burn' chemically - they require vast pressures and temperatures at their cores to initiate nuclear reactions.
James' answer is of course correct.
But we astronomers regularly use terminology like "hydrogen burning" and "helium burning" to describe nuclear fusion reactions in stars, so perhaps we have inadvertently created that misconception.
Our own planet, the Earth has sufficient oxygen to maintain combustion, which it does permanently.
However this combustion is very slow in its natural form and we call it oxidisation, producing things like vinegar, rust, salts, rocks etc. etc. This means that there is a huge amount of combined oxygen (or rather burned matter) in our ecosystem - much more than is in the atmosphere.
A similar but even slower process is maintained with nitrogen which combines with many materials through a burning or oxidation like activity.
Lightning bolts and free fires are natural triggers that make oxidation visible, but it is clear that these processes are quickly extinguished.
As an afterthought. Our oceans are clearly the result of such an episode or rather many such minor episodes. Hydrogen and oxygen combined in the atmosphere in just the right proportions. Something caused a flash (I don't think anyone smoked back then!) and voila! a couple of billion years later we have life on earth.
The present paradigm has the Earth's oceans being partly condensed from the protosolar nebula, and partly delivered by cometary influx.
In both cases, the water is formed through more mundane reactions of hydroxyl and solar protons (on a suitably catayltic surface). The Earth's oxygen-rich atmosphere is thought to post-date the presence of water.
I'll stick with interstellar man getting the shock of his life as he took his first draw on a reefer. Less accurate perhaps but it kept me giggling for a while.