Erroneous paper published and reviewed on MathSciNet
(Disclosure: I review for MathSciNet.)
how much of the responsibility lies on the MathSciNet reviewer who "validated" the proof?
In my view, very little if any, and by using the word "validate" I think you overstate the case. Reviewing for MathSciNet is not meant to be like refereeing, and reviewers are definitely not asked to check the correctness of a paper as they would for a referee report. The purpose of a MathSciNet review is mainly to summarize what the paper contains, so that prospective readers can quickly tell if the contents of the paper are likely to be of interest to them. I don't think you should take the existence of a MathSciNet review as any sort of "seal of approval" on the correctness of the paper.
The reviewer is certainly allowed to mention errors or shortcomings in the paper if they should happen to observe any, but this is not a requirement or an expectation.
And keep in mind that unlike an referee report, MathSciNet reviews are public to the world and signed with the reviewer's name. If a reviewer says something negative about a paper, especially a very high-profile paper, they might face blowback in a way that an anonymous referee wouldn't. Maybe you think they have an obligation to do it anyway and face the consequences, but it's a significant ask.
Useful reading is the Guide for Reviewers that MathSciNet points reviewers to.
My understanding is that it is their duty to reveal the mistakes in the published article, so that the other authors do not base their work on it.
I would only agree insofar as to say that if the reviewer is actually aware of a specific mistake in the paper, they should mention it. Even so, that would only be out of a general civic duty to the community, not any responsibility explicitly laid on them by MathSciNet. And for me, a vague sense of unease that something is not quite right with the paper would not rise to that level. If the reviewer does not happen to find a specific error, I do not think they have a responsibility to go looking for one. Certainly in my time as a reviewer and in talking to others, I've never had an impression that this was expected.
Is there a way to "nudge" the reviewer to make amends to the review?
Their name is attached to the review, so of course you are free to contact them and say something. But as I mentioned, I think what you're asking is beyond the scope of what's expected of a reviewer. If you think the reviewer actually knows of a specific error and has covered it up, that would be another story, but as I said I don't think they have an obligation to carefully check a paper in hopes of finding errors.
Moreover, MathSciNet doesn't really encourage reviewers to revise their reviews after initial submission. There is no automated system to submit revisions. So even if the reviewer wanted to make changes, they'd have to contact the MathSciNet editorial staff and convince them that changes were warranted.
Now some other articles (published in first-tier journals) use this questionable result.
Well, it's the responsibility of those authors to satisfy themselves of the correctness of results that they rely on. They cannot reasonably take the mere existence of a MathSciNet review as positive evidence of the paper's correctness.
how much of the responsibility lies on the MathSciNet reviewer who "validated" the proof?
None. The instructions for reviewers don't ask the reviewer to check the validity of the proofs; by a time a paper gets to MathSciNet it has been peer reviewed already. The purpose of a MathSciNet review is to explain what's in the paper and why someone might want to read it.
It is fine to write to contact AMS and ask them to retract a review, if (1) you are a well-known expert in the field, capable of speaking (to a reasonable extent) for the field as a whole; or (2) you are able to conclusively demonstrate that the paper has an error.
If neither of these is the case, while your desire to do something is admirable, realistically there is probably not any effective action available to you.
So my question is: how much of the responsibility lies on the MathSciNet reviewer who "validated" the proof? My understanding is that it is their duty to reveal the mistakes in the published article, so that the other authors do not base their work on it. Is there a way to "nudge" the reviewer to make amends to the review?
None. Unlike what your title implies MathSciNet reviews are not referee reports nor are they endorsements. As stated in the guidelines, the reviews are there to help some decide if they wish to read the paper.
Incorrect. The primary duty is to provide context to the paper. Moreover, as noted in the instructions, the review is not meant to start a debate as the original author cannot answer.
This assumes or implies that somehow the review motivated citations to the paper, which would be extraordinary.
If you believe you found an error, you should supply evidence of that error in a comment to the original journal or another journal. This is the most constructive way of correcting a wrong paper. An alternative would be to use PubPeer.
Frankly: why the excitement? People cite wrong results - including papers that have been officially withdrawn by journals - all the time. See this table compiled by Retraction Watch of the Top 10 most highly cited retracted papers. #2 on that table has over 1000 citation since the article was retracted.