What methods - inspired by Haavelmo’s Structural Econometrics - can show that a partial equilibrium model is unreliable?
I can think of two big ways.


Formal tests of model fit: If you are using SMM, GMM or indirect influence check out the J-stat. If you are using maximum likelihood you want a likelihood ratio. Bayesian model testing is similar to likelihood ratios, but more complicated. Many of the DSGE models were rejected on these bases (you can see Sargent discussing it briefly in this interview). These tests are all very similar.
Out of sample performance: Estimate the model, then use it to predict out of sample data. This is a tough test, but it is super useful. They aren't so common in economics, but statisticians use them a lot (often times they leave out part of the data and try the model out there). You probably want your model to pass a formal test in sample before you do an out of sample test.


All these tests rely on the same, extremely basic, intuition: If the model is good then it should fit the data well. If the model does not fit the data well it is probably not good.