The question says it all. I'm assuming 12-tone equal temperament.
To give a more familiar example, we hear a minor third and an augmented second differently, for one is a consonance and the other is a dissonance. Similarly, we would hear a perfect fourth and an augmented third differently, if we could indeed construct a harmonic-melodic context in which an augmented third would be heard.
I'm not challenging the conceptual relevance of the distinction between enharmonic intervals in equal temperament, I'm just curious to find out if there are situations in which those specific intervals would arise in a tonal composition, opposed to more radical enharmonics like the doubly augmented octave, which I would bet are impossible to realize in practice.