Can covalently bonded hydrocarbons be influenced by a magnetic field?
Just because there are no unpaired electrons doesn't mean magnetic fields can't influence something. Unpaired electrons tend to lead to strong interactions with magnetic fields (the dioxygen in air has two and is paramagnetic, an effect best seen in the liquid phase where a magnet will attract the cold liquid).

But compounds with paired electrons also interact with magnetic fields, though the effects are usually weak. But the repulsive effect is sometimes strong enough to levitate a diamagnetic substance (see this youtube video showing how strong magnets can levitate plates of graphite which is strongly paramagnetic).

It is unlikely that such effects would be easy to see in gases. But you are talking about combustion and the most likely effect will not be on the neutral gas but on the flame where the gas is not neutral but a plasma of charged particles. Charged particles or molecules will have a much stronger interaction with a magnetic field though it might still require sensitive apparatus to detect how the field makes a difference.