Given the fact the finite population correction effectively reduces the standard error of the sampling distribution, which (compared with the lack of a finite population correction) will increase the test statistic from a hypothesis test (and therefore increase the probability that the null will be rejected), it seems quite a powerful tool to use without a strong justification.
Here is one scenario I feel it is justified to use it:
[Edit 2 - I have changed the scenario here slightly following a very valid point from Steve. I don't want non-response rates to distract from the focus of the finite population correction].
I have a company which has new leadership in the past 12 months. During the old leadership, I performed a random sample on 50% of the employees (with a 100% response rate). Under new leadership, I conducted the same survey on a random sample of 45% of the employees (again, with a 100% response rate)
If I want to see how the results to some questions in the survey have changed in a statistically significant way, I should apply the finite population correction. I am comparing two sets of employees at two specific points in time. I don't care about anyone other than the population working within the company at those two points in time.
[Edit 2 - To clarify my concern]
In addition, those who were not working in the firm at those two points in time would not be able to answer the survey anyway. Taking a basic question like 'I have enjoyed working here in the last couple of months,' only those who have actually worked at the firm could reasonably answer. In which case, assuming an infinite population seems like it is applying an arbitrary restriction on the sampling distribution.
Here is one scenario I feel it wouldn't be justified to use it:
I carry out a questionnaire to a sample of customers who bought products from me to assess their interests, in order for me to inform me how best to diversify my business for the next couple of years. While this sample is 20% of all those customers I have served since I opened, I could perceive the population here to be inclusive of all those customers who haven't bought products from me (because they aren't in the area or my prices were too high or they didn't know about my shop etc.) and therefore I have a population which leaves my sample to be <1% of it in size).
Do people agree with these my interpretation of the two example scenarios?
When producing a piece of work, is it good practice to include the justification why you are using a finite population correction?
Is this largely a assessment of 'who' the population is and whether you want your results to apply just to the 'known' population or whether you are seeking to provide something that has a wider level of application than just the 'known' population?