That's a hard question to answer without importing a lot of bias or metaphysics. I'll try.

tl;dr: Non-locality tends to break causality. Merely relaxing locality is hard (as in nobody who has tried has done it successfully) to reconcile with astrophysical observations (e.g. radiation from periastron advances of binary pulsars, radiation from astrophysical megamasers, JEM-EUSO/Fermi-GLAST observations of gamma and cosmic rays).

A local (relativistic field) theory means that the solution to the field equations is locally Lorentz-invariant and generally covariant. I'll explain these briefly.

Local Lorentz-invariance essentially means evolving systems of matter must obey a fundamental speed limit that applies to all observers and observables. No-one will see matter suddenly appear out of thin vacuum, nor vanish into it without a trace. Everyone can relate certain quantities they observe (e.g. angular momentum), and understand (maybe with some effort) why the observed quantities differ for different observers. We can be rigorous about the immediate "before" and "after" of the arrival of matter at any given choice of here-and-now. This constraint on the fluxes of energy and momentum leads to conservation laws that are for all practical purposes completely accurate in any "small enough" patch of the whole spacetime. (And for particle physicists local Lorentz invariance gives a time-dependent spatial location of a centre of momentum.)

General covariance means that the theory does not require any particular choice of description of "here-and-now". We can solve the field equations if "here-and-now" is at a coordinate origin, and also if one or more of the dimensions is nonzero in some arbitrary system of coordinates. More importantly, the coordinate-dependent quantities in such a solution transform in a precise way as one switches from one system of coordinates to another. It also gives us a clue about what aspects of a system are coordinate-dependent and what are coordinate-independent, and that only the latter are reliably physical.

Locality in this sense gives us the strongest form of causality, which lets us move from solving the field equations for the entire "block universe" spacetime (and then examining pieces of it) to preparing an initial value problem on a single slice of that spacetime, and then extend from there to recovering the entire "block universe". The technical term for this is global hyperbolicity.

In general it is safe to conjecture that as one weakens the causality on a spacetime (i.e., from global hyperbolicity to one of the other choices at <https://en.wikipedia.org/wiki/Causality_conditions#The_hiera...>), the less likely the spacetime is to model any part of our universe with useful accuracy. I'd sloganize it as "anything not forbidden by causality conditions is mandatory in a large enough patch of spacetime". Local Lorentz invariance turns out to forbid a surprisingly large number of possibilities (including lots of weird stuff that we have never seen).

The Standard Model of Particle Physics is a local theory, and it is updated as new experimental and observational evidence piles up. Despite many searches for LIV (Lorentz-invariance violation), it remains fully locally Lorentz-invariant. See <https://en.wikipedia.org/wiki/Standard-Model_Extension> for some details.

I think one should be careful not to confuse locality with local realism. That imports the idea of "scientific realism" in the sense that at a point p there are definite properties which exist -- whether measured or not -- and that those properties can only transform locally as above.

One can get to a non-realistic theory which is still local by (for example) pursuing a (I think pretty metaphysical) claim that "what you don't measure isn't there", or perhaps more weakly, "if you don't measure it, you can't assume the value". Such a theory can still be local without having to philosophize much: "if you do measure a value at spacetime point p, then measure a value at timelike-separated spacetime point p', the values will turn out to be related by a representation of the Lorentz group; but you may not be able to predict p' only from the measurement of p". (The realistic position is that you don't have to measure p' -- by measuring p you already know what p' is. [1])

(The local constraint is interesting in the paragraph above in that it implies that quantities must be combinations of only four-tensors (and four-vectors and scalars) and spinors; c.f. my paragraph beginning with "General covariance").

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[1] Einstein is frequently alleged to have sloganized this as "I like to think that the moon is there even if I’m not looking at it." This comes from a broken-telephone-effect upon A. Pais, Rev. Mod. Phys. 51, 863 (1979), near bottom of first column p.907, <https://doi.org/10.1103/RevModPhys.51.863>, which you can feed into sci hub): 'We often discussed his notions on objective reality. I recall that during one walk Einstein suddenly stopped, turned to me and asked whether I really believed that the moon exists only when I look at it. The rest of this walk was devoted to a discussion of what a physicist should mean by the term "to exist."' I think that's an excellent topic for discussion and that it is not yet resolved.