Just about every approximation, ever. The further you are from math, the more of these there are, and you don't need to go far (all physics other than fundamental physics) to be loaded with them.
Simple harmonic oscillators? In 99% of cases, that's locally true around the minimum.
PV = nRT? Ideal gases are an approximation that is often strong, but right around condensation points or if there are long-ranged forces in the gas, it isn't.
So on, so on.
(all physics other than fundamental physics)
You don't even have to leave fundamental physics. Firstly the existing equations are (presumably) approximations to the underlying unified theory; secondly, we can't solve them exactly anyway, and even in the relatively tractable electroweak case we use the approximation of a truncated sum. As for the strong force, where that technique doesn't converge, don't even ask.
New Salt Compounds Challenge the Foundation of Chemistry
The title is overblown (it depends on what you think the foundation is), but get a load of this:
And here's the philosophical bit:
The obvious example of local truth is relativistic effects being pretty much invisible over the durations and distances that are normal for people, but there's also that the surface of the earth is near enough to flat for many human purposes.
Any suggestions for other truths which could turn out to be local?