Would a satyr wear horseshoes?
Horseshoes were invented because stone roads were bad for a horse’s hooves. I’m not sure if a satyr would want to nail horseshoes (goatshoes?) to their hooves, or if they would come up with some way to put shoe-shoes on hooves.
A horse’s hooves are monodactyl; single-toed. This means they consist of a single large digit. 
The hooves of a goat (or other ruminant) however, are cloven: they consist of two digits, and are shaped very differently to the hooves of a horse.
Even if they wanted to, a satyr (which has the legs of a goat) could no more wear horseshoes than you could wear a pair of dog booties. (Which are adorable, but I’m sidetracking myself.)
Besides which, a satyr doesn’t actually need any protection from cobbled roads. Horses are built for the soft ground of open grasslands, so it makes sense for them to wear horseshoes when on hard ground. But goats are mountain-dwelling animals, and their feet are probably tough enough to tolerate cobbles.
My bet: Yes. Or something else with the same end goal: protecting the hoof from excessive wear.
I see two factors in play: too much wear on the hoof and satyrs are bipeds.
Wear: Hooves (whether for horses or goats) grow naturally all the time, and there is a balance between normal wear and normal growth that tends to work out in the wild. Domesticated horses -- or even oxen put to work -- may get shoes because they are wearing their hooves down from overuse on hard surfaces. If your horse is pounding on pavement or packed trails regularly, it will need shoes. Conversely, you can need to trim hooves of farm animals if their hooves are underused. Pet goats usually need hoof trimmings as they are not getting the normal wear and the hoof just keeps growing.
With all that in mind, my bet is that a satyr living in civilization probably will want hoof protection, assuming they are spending most of their time on packed surfaces -- inside their homes, walking on roads, etc. If your satyrs live in the wild and eschew modern conveniences like paved pathways then perhaps they don't need them, but....
Bipeds: Goats, cows, horses, etc, distribute their weight across 4 hooves, versus a satyr distributing the weight across two hooves. My thinking is they must necessarily have more wear and tear, then. Perhaps this means satyr hooves grow faster than horse (or goat) hooves but it may also make them more susceptible to damage, especially on paved surfaces.
Their shoeing options should be a lot more open. The important thing with a shoe is that it protects the hoof from wear. Satyrs presumably being intelligent creatures capable of tying their own shoelaces could potentially wear something much more like a conventional human shoe, with a soft sole, and then go in for, erm, whatever you would call a pedicure for a satyr, to keep the hoof trimmed. I suspect that they can do cloven hoof shoes that nail in (see: oxen shoes) but a civilized modern satyr could do better.
The details are entirely up to you but hopefully this helps you nail down the idea.
I upvoted @JamieB's answer, and you should too, but let's build on it.
There are humans who pretty much live their entire lives without ever wearing a pair of shoes, using a pair of gloves, or getting their teeth capped. Some don't wear hats, or even much clothing. Heck, some don't wear clothes. There's nothing actually wrong with this. Human are born with everything we need to live life — given the correct environmental conditions.
While humans may choose to live in areas where they must wear, e.g., coats and boots, the truth is, we wear all those things (including the tooth caps) to protect ourselves and/or extend our ability to do things.
Let's ignore the fact that your question depends too much on narrative necessity
Frankly, there is no more of a reason why satyrs would want to wear horse shoes than there is one explaining why humans would want to wear athletic cleats, ski boots, or work boots. Whether they do or not is up to you because, like humans, they don't have to. In fact, with some exceptions, a hoof wears as it is used. Regularly shoeing hooves means the need to trim those proverbial nails, too, because they're not getting worn as nature intended. But why would a satyr want to do this?
Good grief. Why wouldn't they?
Combat: the shoe not only protects the hoof but is better than wearing a gauntlet or holding a roll of quarters in your fist when you pummel the snot out of someone.
Labor: The shoe gives the satyr the ability to push forward with greater strength than the hoof itself might allow. Or, at the very least, allow the satyr to grip or dig into the ground better than the unshoed hoof would.
Dancing: I have one word for you, just one word... tap.
Injury: Like a cast, the shoe wold hold the hoof together while healing takes place.
Vanity: Why metal? Why not rubber or plastic or wood or any number of things that changes the appearance of the satyr? Platform shoes, decorative shoes, clogging shoes (ah, I've already mentioned dancing...). Maybe it's because your wealthy satyr wants to show of his/her status by wearing their outrageously expensive African Blackwood shoes.
And once they start wearing shoes, time (if we use Human history as the reference) will lead to pretty much all of them wearing shoes. No shirts, no shoes, no service. It'll become the essence of polite society to wear shoes so they can be removed (ah, the need to invent removable shoes!) when you enter someone's home — because tromping mud around someone's house is simply gauche.
Of course they'll wear shoes...
It's actually really difficult to rationalize why an entire species of intelligent, occasionally selfish, ocassionally noble, ocassionally just eccentric beings would or wouldn't do anything. There is nothing, absolutely nothing, that holistically or rationally allows this. In other words, we're back to narrative necessity.
But there are plenty of reasons (as many as for humans, frankly), why a satyr (note, one) would choose to use shoes.
Now that I write that last sentence, please note that the Help Center specifically prohibits questions about character and/or organizational choices. Hold that thought for future quesitons. Technically, now that I'm done writing an answer, this question was off-topic.
Shoeing has many benefits, reducing wear on hard surfaces, softer shoes reduce shock to joints, they can correct foot problems. Remember not all of a hoof is equally hard, thee are softer parts that benefit form protection. Then you have things like combat, protecting the rest of the foot, and even just vanity.
Historically they might use something like OX-shoes which nail on but I expect glue or tie up shoes to quickly win out, see roman hipposandals (image below) for iron tie up shoes, these had a woven pad on top and a metal bottom. Hipposandals would be harder to make for cloven hooves, but they might just be one piece since you can take them off. Entirely woven hipposandals also existed. I expect hipposandals to be common. We don't make shoes for each toe and I don't expect them to do so often. Hipposandals actually come in a huge variety of designs including cleated, quick removal, soft bottomed, and highly decorative.
I expect tie up or lace on shoes more like boots for the wealthy. Shoes you can take off make more sense for an intelligent being but would also cost a lot more. there is a reason wooden shoes for so common for the poor though most of history. But maybe the rich use a prettier metal, that's up to you, different societies would likely be different. Few things show off wealth better than using silver for a shoe, especially given how fast it will wear.
Wooden ox shoes also existed, and were glued on and are still used to protect hooves form hard surfaces for a short time. These might be your most common especially for poor satyrs. My guess is switching to glue on wood or iron shoes would quickly win out over less comfortable nail on shoes if they even try nail on shoes, and I don't imagine they would be more common than hipposandals.
in a more modern setting they would likely move to only glue on or tie up shoes which is what is mostly used now, since there is far less chance of splitting a hoof. These can be made of everything from iron to plastic to soft rubber. today these are called cowslips or hoof blocks, and even orthopedic varieties exist.
