It depends.  Your question is a good one.  The answer is not so much that the IRS knows that your rental is a "not for profit" activity, but rather that it would be easy for them to tell that it is not a "for profit" activity if you tried to report it as such.

Here's the idea:  the IRS may not know from you putting the income on line 21 that the rental income is not a fair-rental value for the dwelling you are renting.  However, if you were to report the rent you receive on Schedule E, and then the expenses, they could easily cross-reference the home with the Fair Rental Value on similar dwellings in your area and be able to tell immediately that you have a "not-for-profit" rental.

And, since you can't claim expenses (or QBI deduction either) against a not-for-profit rental on Line 21, it is extremely unlikely that they are going to question what is reported there.  (That doesn't mean that they can't question it, though).

A "for-profit" activity does not actually have to have a gain to be considered a "for-profit" activity, because, even at a fair rental value, there are a number of reasons a rental activity could produce repeated losses.  However, many of these losses are recaptured when the rental activity is eventually sold.  But if you are renting below fair rental value, you are correct in not reporting the rental on Schedule E but rather on Line 21, Schedule 1.