I played a Prince and set it aside so that it would be played at the beginning of every turn. After a few turns I no longer wanted to set it aside anymore. The card text says to stop playing the Prince when you fail to set it aside, but I could not find a way to stop setting it aside. Did I miss something? Or is this a bug?
This sentence is not giving you an option to stop playing the card set aside with Prince, it is telling you a circumstance under which Prince stops "working" (that means it no longer plays the set aside card each turn). It's (in 99.9% of cases) a disadvantage.
The easiest example are Duration cards; they are (usually) not discarded at the end of the turn they are played, so Prince fails to set them aside and so, by the sentence you gave, will not play them any more in future turns.
Other examples are cards that can trash themselves, like Mining Village, or that move themselves elswhere (say to the Supply with Way of the Butterfly).