Theorists hate it and empiricists love it. Let's talk theory.
I think you can do a lot of damage by maintaining strong biases against exogeneity in a model as an economist. We have primarily economic explanations for phenomena. Sometimes things that are left exogenous ought to be endogenized because an economic argument can do the work of endogenizing them. A classic example is the savings rate or the marginal propensity to consume (although even these ultimately boil down to some deeper endogenous parameter - but the exercise of fleshing savings behavior out a little more is still informative).
But I am becoming increasingly uncomfortable with other attempts at endogenization, seemingly for the sake of endogenization itself. If we don't think things are economically determined, we are probably safer leaving them exogenous. That doesn't mean we can't explore elsewhere for clues to the origins of these other things, of course.