Headline grabbing effect sizes with sexy variables in complicated models are, over the long run, far more likely to be found wanting than vindicated as possibly right. Science progresses with small, incremental contributions to our knowledge base.
Weak instruments are a problem in any method dealing with endogeneity where an instrument varible is a proxy for random selection. Heckman selection models share a similar problem of weak instruments, and it has to do with the exclusion restriction.
In the case of logit models with continuous predictors,it takes some extra work to make sense of and really get a handle on the relationship the predictors and the outcome. Marginal effects and predicted probabilities are, to me, a must have in logit model analysis, particularly with continuous predictors.
The bottom line is that there is no substitute for using your own judgement when evaluating a study. Ask yourself just how likely it is that the null hypothesis is to be true, particularly when evaluating research purporting to offer surprising, novel, and counterintuitive findings.
I am actually a big fan of theory; I amm just not wild about the ways in which we (management and entrepreneurship scholars) test it. The driving reason is theoretical looseness; the ability to offer any number of theoretical explanations for a phenomenon of interest.