Although you target Iona as you cast Unburial Rites (C.R. 601.2c), so that all players will know what the target is before that spell resolves, you don't choose a color for Iona until Iona enters the battlefield (e.g., due to Unburial Rites resolving) (C.R. 608.2, 608.2c-d). However, Iona's effect that "[y]our opponents can't cast spells of the chosen color" starts the moment it enters the battlefield, so that, for example, if white was chosen for Iona, Iona's controller's opponents can't cast white spells (including Path to Exile) while Iona is on the battlefield (C.R. 611.3b, 112.6, 109.5). (See also Llawan, Cephalid Empress.)
Note that Path to Exile can target only a "creature", not a creature card in a graveyard, so that Iona in the graveyard can't be dealt with using Path to Exile (C.R. 109.2, 114.1a).
Note that Path to Exile can target only a "creature", not a creature card in a graveyard, so that Iona in the graveyard can't be dealt with using Path to Exile (C.R. 109.2, 114.1a).
EDIT (Jun. 17): Correctness edit.