A lot of people assume that it's just the way that coins smell, and the odor is rubbing off on their hands, but you're not smelling the metal so much as you're smelling yourself. That funky scent is actually a human body odor created by the reaction of oils in the skin contact with objects that contain iron (a separate, but similar odor, is created when we touch copper). What we think of as a "metallic smell" is only metallic by association.
Here's how it works.
When you touch something made of iron, perspiration on your skin cause the iron atoms to gain two electrons, and these doubly negative iron atoms react with oils in the skin, forming several types of compounds called aldehydes and ketones. If you've gotten a whiff of formaldehyde while dissecting frogs in science class or wrinkled your nose at the smell of acetone in nail polish remover, then you know that these compounds have very strong, distinctive odors.
When German researchers captured and studied the chemical composition of vapors coming from the skin of people who had just handled iron objects, they found one compound, 1-octen-3-one, that has a particularly pungent fungus-meets-metal smell and is detectable by humans at very low concentrations. This compound may be the main component of the odor of your hands post-money-handling.
You might have noticed that a similar smell is produced when blood meets skin. That's because blood contains iron and these iron atoms go through the same reaction you're your skin as the atoms from coins, producing the same scent molecules.