No, unless you can add a fire/acid descriptor to the spell
Burning Spell clearly says that the spell must have the acid or fire descriptor, however, there are ways to change a spell descriptor to include acid/fire.
Such as the Admixture Wizard's Arcane School:
Versatile Evocation (Su): When you cast an evocation spell that does acid, cold, electricity, or fire damage, you may change the damage dealt to one of the other four energy types. This changes the descriptor of the spell to match the new energy type. Any non-damaging effects remain unchanged unless the new energy type invalidates them (an ice storm that deals fire damage might still provide a penalty on Perception checks due to smoke, but it would not create difficult terrain). Such effects are subject to GM discretion. You can use this ability a number of times per day equal to 3 + your Intelligence modifier.
The Sorcerer Bloodlines that deal with elemental damage, such as Elemental, Efreeti and Shaitan bloodlines, all got an ability that adds fire or acid damage to the spell descriptor, such as:
Bloodline Arcana: Whenever you cast a spell that deals energy damage, you can change the type of damage to fire. This also changes the spell’s descriptors to match this energy type.
The Elemental Matamagic Rod however does not change the spell's type, as it works like Elemental Spell feat, which doesn't include that specific ruling.
The Bloatmage can also have the same powers as those sorcerer bloodlines mentioned before, using their 10th level ability Absorb Bloodline.
And finally, this is really stretching it, but seems to be legal from a rules a written standpoint:
All Summon Monster spells seems to be valid targets for Burning Spell.
When you use a summoning spell to summon a creature with an alignment or elemental subtype, it is a spell of that type.