Not into a higher cabin class and increasingly not into more desirable seats such as front or exit row. Forward seats are sometimes sold at a premium so its worth checking when booking if moving from 7C into 3C would involve a cost.
Often the premium seats are distinctly marked as such with headrest covers or different trim and the cabin crew will make an announcement offering upgraded seats available for purchase.
A shift across the aisle into an empty row is generally okay however, sometimes seats are allocated for crew rest so it never hurts to check.