Am I morally obliged to report what I judge as self-plagiarism?
This is an interesting case that you cite. I think it falls somewhere between 'standard behavior' and 'clear misconduct' (though I am sure others will disagree). Some thoughts and facts:


In Computer Science, even though both journals and conf proceedings count as papers, it is common accepted practice to submit extended versions of proceedings papers to journals. Typically, journals expect between 30% and 50% new content for such papers.
Realistically, many journals (especially ones that are not exactly top quality) are not super-strict with what they consider "additional content". Describing only their methodology in more detail, without providing new insights or results, is certainly very little new contribution, but I am not sure if that already qualifies as misconduct.
That one of the original authors was missing in the journal version is curious, but there may be reasons for that. Presumably, something a little shady has happened along the lines, but it is impossible to tell from the outside. One realistic possibility is that first author A has submitted the conf paper while at institution (1), with help from B and C. D was the supervisor and contributed little to nothing at all, but was added due to lab policy. Now, before submitting the journal version, A moved to institution (2), and has replaced lab head D with lab head E.
What strikes me as very, very odd is that the journal paper does not make clear that it is an extension of an earlier conference paper by citing it (and saying so explicitly in the introduction). This is the one thing that I certainly agree has a very bad smell to it. If I receive such a "hidden" journal extension for review, it is an auto-reject for me. 



  Now, am I morally obliged to contact the editor of the journal where the second paper was published? This is not precisely a double publication, nor are the authors damaging me nor anybody else. Still, I feel they did something wrong. Should I instead live and let die?


In summary, I would say the entire case certainly does not look good. That being said, I do not think that the case is blatant enough that you need to feel required to report to the journal editor. To be honest, even if you did, I think not much would happen. Likely, it is not a paper that the journal feels overly proud of, but I do not think that the case is bad enough to warrant formally retracting the paper.