Yes, it happens. There are plenty of examples out there, in serious and pop music. The finish on relative minor is, as Patrx2 stated, the deceptive, or interrupted cadence, leaving the listener with a feeling of being left hanging on for the end, which has actually just happened.
The 'tierce de Picardie' where the last chord is 'borrowed' from the parallel major is far more common than the change you suggest, but playing a whole piece in major, and finishing on the tonic minor is rare. It would leave the listener on a down, unless it was setting up for the next movement, or even the middle 8 of a pop song, but isn't often used. Best to try each and see - or better still, listen!