Back in I think it was the 70s there was a journal article titled "Uncle Remus and the Cascade Effect in Clinical Medicine: Br'er Rabbit Kicks the Tar-Baby". In the classic story of Br'er Rabbit and the Tar-Baby, the other animals make a figure out of tar to play a joke. Br'er Rabbit approaches the figure and is annoyed when it does not answer his greeting. Eventually, he decides to teach the figure a lesson, and punches it. Of course, he gets stuck in the tar. The more he attempts to fight it, the deeper he is sucked in to the sticky tar. The author compares this to a care provider doing all sorts of tests for the simple reason that it is possible to do them. The more tests you do, the more likely you are to find something wrong - especially because with the way the tests are designed, a small number of healthy people are going to have an abnormal result. Of course, being a professional, if you find something abnormal, you're going to want to do something about it. The concern, though, is that, in a manner reminiscent of Br'er Rabbit, the provider ends up going further and further down a path involving increasingly more invasive procedures and additional testing - for a problem that may well have questionable clinical significance. The example the author gives in the article is that it's not uncommon for people to have a benign mass on their adrenal gland. In some cases, this mass may be a pheochromocytoma, which is a benign tumour that secretes adrenaline. This can be dangerous because it can cause an increase in heart rate and blood pressure that can put stress on the heart. In truth, though, while a benign adrenal mass is common, a pheochromocytoma is rare, so more than likely the mass is not significant - but of course, because you went looking for it and you found it, you now feel obligated to attempt to fix it.
Another example, one that wasn't mentioned in the article but is something I have heard of, is the age at which a smear test is recommended. It used to be the case that it was recommended for every woman at 18 years old, even if she had never had sex. Subsequently, it was discovered that in many cases of HPV with related cervical changes, if they're occuring in women under 21 years old, the body will typically clear the virus and any associated cell changes within 3 years. No need to do anything. But of course, if you found something, you'd want to address it - and depending on what you found, it could require more frequent monitoring, colposcopy, or even cervical biopsy. At best, this could cause unnecessary worry and unnecessary procedures with associated costs. At worst, if biopsies were taken, it can potentially affect a woman's ability to carry a pregnancy to term, because biopsies can thin the cervix and make it more likely to open too early. They subsequently increased the age at first screening (I think to 25 or something), and indicated that it should only be done at least three years after first intimate sexual contact, whichever is later.
Screening is all about balancing the risk of missing something important and the risk of finding something of questionable clinical significance, which you then feel obligated to address (with associated costs and risks to the patient).