Whether coins or banknotes, collecting from circulation is almost always about "interesting" and not "valuable." I've personally never gotten a BU
Morgan dollar in change at a restaurant, so you've got to keep those same expectations in check with banknotes. If however, you wanted to collect a modern notes from every district, then that's certainly something you could work on, especially if you aren't fixated on getting uncirculated notes.
Also, while I know some people don't mind spending hundreds of hours looking through boxes of cents to find a handful worth a few dollars, I'm not sure how many people get $10,000 in 1-dollar bills to go through on a regular basis. If you find fancy serial numbers or star notes from limited runs, definitely pull them out. Personally, the closest thing I've found to "valuable" in circulation was a 1957 $1 silver certificate--worth approximately $2.
I do pull out uncirculated notes occasionally to have an example, but not expecting they will ever increase in value. Inflation obviously lowers the value in real terms over time. Of course, the same can be said for banknotes pulled out of circulation a hundred years ago. Those notes didn't start off being worth more than face value.