If a paper is rejected by many journals before finally being published, will that cause conflict with the reviewers who rejected the manuscript?
Your referees at specialist journals said, basically, "These methods aren't good enough to answer these questions, because X." But the generalist high-impact journal may have looked more at the results than the methods.

If X is "we just don't think this is very sophisticated, and the tools are out of fashion," I suspect nothing will happen.

If X is "there are uncontrolled errors larger than your effect," and you haven't explained in your paper why that isn't true, someone might write a comment on your paper. This comment will look a lot like your earlier referee reports. The field will then decide who is right. (See, e.g. the "arsenic life" experiment for a recent high-profile example.)

Beyond this, if I saw the second case from a paper I refereed, and there was no attempt from the authors to address the points I raised, I would not be happy with the authors, and might be less likely to extend them the benefit of the doubt in future referee reports.