In the US, very little. That's because there really aren't many meaningful protests these days, because the US has effectively killed the public spaces protests rely on. But the Tea Party movement was arguably influential, and protests were its main tool. Occupy Wall Street was also far more successful than commonly believed: the 1%-vs-99% divide has become a central narrative of US politics, even on the right.
Outside the US? The Arab Spring was nothing but protests, and it did topple something like a dozen governments. Yes, the new ones mostly failed, but that's a different matter.
In Ukraine the protests succeeded. In Hong Kong, they certainly seem to be a major pain for the local government as well as China.
Environmental protests have succeeded almost spectacularly over the last decade or so. Germany's anti-nuclear movement relied on protests as their main method-the Green party was actually founded by a group of people who first met at a protest. Today, Extinction Rebellion, Fridays for Future and similar keep climate change in the news week by week. That's the sort of pressure without which the Paris accord would have never been attempted.
Compare any agenda from a G7/G8/G20 meeting a decade or two ago, and you'll notice that issues such as the environment, labor protection, or fair trade have gone from zero to dominating these meetings.
The US-EU trade agreement basically died on the day that half a million people in Berlin protested against it.
Obviously, it is often impossible to unambiguously assign causality. It's a complex system where everything affects everything.
But purely subjectively, I am actually surprised by how effective protests are. Consider Ukraine, where a single protest at a central square toppled the government. In practical terms. a Tiannanmen-style massacre was entirely possible. But the protesters won. Why? How?