Bill Greene visited my school (UConn) last year. We asked him about, whether it was essential to have exclusion restrictions in a Heckman selection model. He said no, that the non-linearity alone is sufficient for identification. When he got some push back, he recounted an anecdote where he was one of three reviewers on a paper where this was the central issue, and ended up losing the argument to the other two. The quote he ended with that stands out is, "I am comfortable in the non-linear world."
The next semester we had one of your students visit (she was working in Corporate Finance at the World Bank). She had a Heckman model in her paper under robustness checks, and I asked her why she didn't just make it her main model and sidestep some of the endogeneity issues that arose in the talk. She said that, since it didn't have exclusion restrictions that it was bad form. She said that Heckman wouldn't like it and Heckman's students wouldn't like it (I guess that includes you, right?).
I guess what strikes me most is that people at the top of the profession disagree on this issue. It doesn't strike me as an open question, so much as different people knowing all about the trade-offs, coming to different conclusions (and maybe different conclusions about the right "rules of thumb").
So if you think I've summarized the views correctly, I think a post your position and why you and Greene disagree would be interesting.
I don't think Bill and I actually disagree much. We certainly agree that, in the technical sense, no exclusion restriction is required to identify the bivariate normal selection model. That is a non-controversial matter of technical econometrics. Where it seems we might disagree is whether reporting estimates based on the bivariate normal selection model without an exclusion restriction adds any value. I would say it does not add any value, and actually subtracts value in some sense by potentially misleading econometrically uninformed readers.
I suppose that the counter-argument would be that it is better to do something than nothing, where something is the bivariate normal model without an exclusion restriction. That's fine, but I would choose a different something. In particular, I would much rather see a sensitivity analysis that fixes the rho parameter (the correlation between the unobserved components of the selection and outcome equations) in the selection model at different values and shows how it affects the estimates than estimates of rho without an exclusion restriction. The sensitivity analysis should, in my view, be accompanied by substantive arguments that limit the reasonable support of rho in the particular context under study. [The related versions of sensitivity analysis from the matching literature - see e.g. Ichino, Mealli and Nannicini (2008) Journal of Applied Econometrics - would be fine with me as well.] The sensitivity analysis approach seems much more honest about the nature of the evidence than simply reporting "selection corrected" estimates based solely off of functional form assumptions.