I'm a first-year graduate student studying linear regression.

Why is it appropriate to use a linear model when we know that real life relationships are never perfectly linear? It seems like often I have some cloud of data and I fit a linear model, and often I get a significant coefficient. But I'm not sure this tells me anything scientifically. The proofs we do in class don't apply in this case.

Is there any educational paper/book that discusses this issue? Can the assumption of linearity be weakened to "approximate linearity"?

I'll give the standard econ answer: It depends.

Firstly, you can indeed have "approximate linearity" as in "although this isn't truly linear, linearity can provide an OK description of E(y|X)".

Secondly, you may still prefer to use a linear model even if this isn't the case simply because it's easier to explain whatever your model is doing (I know, this isn't necessarily a great reason but it's important in practice).

At last, there are ways to check if you should transform your predictors - the classical way is to use various residual plots, a nonparametric way is to fit a kernel regression or an additive model if you have more than one independent variable. After transforming your regressors, you can just do OLS using the resulting transformed design matrix as usual.

Unfortunately, for some reason this isn't usually taught in econometrics courses, at least not the core ones even if it should.