I work with some students who will attend graduate school in the US. The schools they are applying for list minimum required TOEFL scores. Some have just reached the minimum scores required for admission to their schools of choice. Do admissions departments ever care about how close this number is to the minimum? In other words, will the students be at any greater advantage if they retake the test and get a higher score, or is the "score level met" simply a "met" or "not met" variable that goes into deciding whether or not the students are accepted?
The answer is "sometimes."
Some schools will not care what the score is, as long as you have the minimum requirement that they advertise publicly.
Some schools won't care, even if you don't have the minimum requirement (they may "waive" this requirement routinely).
Some schools will care: they may have one public requirement, and a higher (unpublicized) threshold that they really use in most cases. (Or: the school has one minimum and the department has a higher, unpublicized one.)
Some schools will care: the minimum is required to get your application looked at, but even if it is met they still consider the score in admissions decisions.
In addition to other useful remarks made: in my university, a minimum TOEFL score is considered a problem, to the extent that it drags down everything of the file. Partly this is based on our past experiences with such scores (statistically, over 30+ years) and the subsequent improvements-or-not. Specifically, minimal TOEFL score people seem not to usually improve much at all, even after a few years. Oof. Of course this says nothing about any individual, but we predict based on the statistical aggregate...
The minimum set by the umbrella "graduate school" body that imagines it oversees graduate education and degrees across the university has lower expectations than our math dept. I cannot speak for other depts.
And, indeed, both higher and lower TOEFL-score kids who associate almost entirely with their home-language population outside of teaching and classes simply do not improve, do not learn finer points of colloquial English, etc. On one hand, I'm somewhat sympathetic to cultural dislocation... but if the issue is job performance, then we can see the obvious issue: cultural extraterrestrials are rarely effective in teaching.
Almost 20 years ago, I was in charge of graduate admissions for my department, and TOEFL scores were among my biggest headaches. They just didn't provide the information I really needed: Can the applicant succeed not only as a student but also as a TA in first-year calculus or pre-calculus? (TA positions were essentially the only financial support we could offer foreign graduate students.) I remember two students in particular. One had the highest TOEFL score I had ever seen, but I had great difficulty understanding her, and I'm confident that I can understand "foreign English" significantly better than our calculus students. The other student had a TOEFL score barely above the university's minimum, and she was admitted only because one of our faculty vouched for the reliability of a professor who had worked with this student and vouched for her ability to speak English well. This student was, despite her TOEFL score, indeed able to speak English well, and she even served as an intermediary between some of the department staff and some of her fellow students whose English was not so good (though their TOEFL scores were better than hers).
My department also requires applicants to provide a letter from a native speaker of English, attesting to the applicant's ability to teach. (We sometimes stretch the meaning of "native" a bit, for example to include me.) These letters are often more useful than TOEFL scores, though sometimes the language of the letter sounds so foreign that I get suspicious.
Returning to the actual question, unfortunately what departments do with TOEFL scores depends not only on the particular university and the particular department, but even on the particular person or committee that happens to be handling graduate applications. I can easily imagine that my successor as associate chair for graduate studies treated TOEFL scores differently than I did and thereby avoided some headaches).
What ff524 said, but I will add that if you are interviewed they will judge you by that. It's reasonable since if you have the TOEFL with good marks but have already forgotten half your English, you are deemed unreliable, if you failed it on the other hand, but are at ease with English, it is easy to understand that you probably failed the essay or had a bad day. What they see with their own eyes is more important than an exam taken some time ago under who knows what circumstances.
I would agree with "sometimes." Usually departments will set these criteria, not schools, based on the success of students who have a given score. But once that bottom is set you meet it or you don't.