Hi there, this thread asks commenters to compare a historical situation to a new one -- and discussions of Uber itself certainly fall within the limits of /r/AskHistorians' 20-year-rule, and are thus off-limits.
If you're coming here to comment in this thread, we thus ask that you confine your comments to the historical question being asked. This is not the place to talk about your our your city's/town's experience with Uber, Lyft, etc.
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I posted about the luddites, which was a somewhat analogous situation in the 1800s (ish) in England in a lower-level response but here's a top-level response to your exact question, followed by some marginal speculation on why that situation can't be used compositionally in the uber scenario (or the much much much more disruptive scenario to come: self-piloting transport, which is an area of my study):
No. Or more to the point, I've spent many hours looking and have not found any meaningful incident. Absence of evidence, etc. etc.
The first iteration of what we think of as a yellow cab (as opposed to a hired horse/team and carriage) showed up around 1900 in European centers and NYC. There was a workforce of carriage and coach drivers, but mobility into driving self-powered transport was fairly lubricated.
[Discussion points / speculation] The transition from meat motors to steam and dinosaurs was incredibly gradual (compared to the Uber scenario) and the skills/expertise (navigating a city, and its streets -- which were nothing like the "orderly" thoroughfares of today) were readily transferable. Moreover, if we include WWII, the transition to a mechanized society occurred during three periods of profound demographic change (wholesale slaughter of millions of working-age males, and the spanish flu in 1918), which has been shown to increase labor power through supply destruction. If for some reason you couldn't make the transition from coach driving to automobile driving, you could still do literally anything else and generally expect better wages than you had in the past. The depression being its own animal, but separate.
By comparison, Uber is an immediate and profoundly disrupting influence in any market it hits. It's made more so in markets like Paris and NYC, where the right to participate in supply is mediated by a medallion system; essentially you have a huge capital investment in the cartel. I haven't found whether there were similar medallion-type supply restrictions in the past, and I'd be fascinated if someone could dig up whether supply was limited in a similar way during the hackney era. Of course, even owning (or renting) a coach and team was a capital expense of its own back in the day, but you also had longer timeframes for them to depreciate (die and fall apart) during the transition to mechanized transport. So, eventually, a driver or owner would have to make the decision "do I replace this horse, or do I get an automobile".
To make some educated guesses about what the future will look like based on history, your best bet is to look at the history of agricultural/textile labor revolts in the face of profound and rapid technological (as opposed to structural) unemployment.
Quick edit to say that in most american markets it's extraordinarily rare for an operator to own their medallion; they lease it at pretty extravagant cost daily. Which may play into why fewer operators are in revolt stateside, since they can just lease a car and install an app and make the transition to the new economy pretty trivially.
I don't have an answer but I'd like to expand on the OPs question, if that's allowed. What about the people that worked in 'horse support industries', for example blacksmiths, farriers, vets, stable boys etc.? Did they riot? Has an unemployed veterinarian ever firebombed a mechanic's shop?
I would like to mention a somewhat analogous situation with regards to hauling or towing boats upstream which had been a considerable source of employment for a very long time. The first steam powered replacements for people towing such barges were chain ships that pulled an entire column of these upstream along a chain running at the bottom of the river.
In Germany such a chain was first introduced on the river Elbe around Dresden in 1866. A contemporary report mentions that haulers witnessing the very first attempt made derisive remarks at first, believing that the ships couldn't manage the more dangerous parts of the river, but after seeing that the entire column had arrived safely in Dresden they began to throw stones in typical Luddite fashion.
Steam-powered vehicles on rails were quite a thing by around 1830, but powered vehicles on roads were not much until much later. How many countries or states had laws against steam wagons and such on roads? If laws didn't prevent motorized vehicles on roads, what did?
Thames watermen protested against the introduction of licensed carriages for hire, even though the river obviously couldn't go everywhere in London. The most notable complaining about the hackney coaches was John Taylor (1580–1653), who did it in verse.
The hackney business progressed, from driving second-hand private coaches to the design of carriages specifically for the trade, notably the Hansom cab. ("Cab" short for cabriolet.")
Automobiles were introduced rather slowly to city streets. It wasn't like they suddenly cleared the streets of horses one week and brought in autos. You had the horse-cars changed to overhead electric power. Then private cars started showing up in a trickle (in the 1890s, as I once researched, there were no autos produced in the USA). If you look at photos of New York of London as late as the 1920s, you will see, outside the best neighborhoods, plenty of horse-drawn traffic, especially wagons.
When a change takes place over thirty years, there's no need for protests. The change to horseless technology did not deprive anyone of their livelihood, did not hand the field over to amateurs. Horse, steam, electric, or gasoline powered, you still needed a hack license to pick up passengers for hire. Younger men simply started out in autos. Older drivers sold the horse to some farmer and converted when they felt like it. Old men provided service for old passengers who didn't like to drive in machines, or moved into light delivery services. There became a push in the Twenties to convert to autos, but that only changed the means of driving, not who was in the business.
