A living creature composed entirely from one classical element is, of course, impossible under everything we know about biology. However, let's make things interesting and try to determine a way for an organism to look and behave like an elemental, without actually being one. In order from easiest to hardest.
This is pretty easy, actually, since the creature doesn't need to be made of earth, it just needs to be covered in it. There is a kind of crab which goes around collecting rocks, plants, bits of shell, and so on, and uses an adhesive to attach stuff to its shell for purposes of camouflage. Looking like the land around you is a good camouflage strategy, so any creature that values stealth or defense over speed and agility could benefit from this.
It could be argued that most life on earth are already water elementals, since cells are basically little bags of water encased in thin cell membranes. Creatures like jellyfish can be practically invisible if their refraction index is close to that of the water they live in. But for a creature to be truly water-elemental-ish, you probably want something like a slime mold - a microorganism that is capable of joining together with others of its species to form a mobile blob-like creature.
I would also bring up hagfish, a creature that excretes microfilaments in order to convert large quantities of water into a thick slime. A single hagfish can produce gallons of slime this way, more than it would be capable of storing in its body in liquid form.
Imagine this: A population of microorganisms inhabit a large lake, strangely devoid of living fish. When someone comes into the water for a swim, they hardly notice that the water seems strangely sticky, but pay it no mind. As they approach the center of the lake, suddenly they feel the water around them turn to thick, clinging slime...
After eating their drowned victim and reproducing, a number of these microorganisms form a large, translucent blob, which crawls out of the water and makes its way toward another nearby lake, to repeat the cycle.
Microbes can't really control themselves in air all that well, so you'd need something that is too small for the eye to see, yet big enough to resist normal brownian motion. Massive swarms of very, very small insects (the smallest known flying insect is a fairyfly, about a tenth of a millimeter in length) with flocking behavior, perhaps, seeking to mate or hunt... but they probably wouldn't feel much like wind if they hit you, and the thicker the swarms, the less likely they would be to be invisible, and as it got close, you'd be able to see it for the swarm of insects that it is. From a distance, it would end up looking more like a (slow-moving) mobile smoke or fog.
On the other hand, if we're talking about air elementals, as in, air which can attack you... well, that's just airborne diseases, isn't it?
For a creature to constantly burn, it would require a lot more energy than it could ever hope to acquire through this method, and it is hard to think of a reason why this would be useful. Instead, let us consider the possibilities of a microorganism that starts fires.
Eucalyptus trees are fire-resistant, and drive out competing plants by producing flammable oils and waiting for something to light them on fire. Imagine a microorganism, perhaps descended from volcano-dwelling, heat-loving archeans, that has learned to colonize cooler regions by spreading spores which chemically ignite dry wood on contact in order to drive out competing organisms. Unlikely, sure, but we're talking about scientifically plausible fire elementals, so I think we can stretch probability a bit here.