Can you just exhaust Irelia without moving her?

No — you can't just exhaust Irelia without moving her.

Exhausting is a Limited Action that only happens as a cost for an action/ability or when a card effect tells you to (Rule 414.3). The Standard Move is the only generic exhaust you have, and there "exhaust" is the cost while actually moving is the effect — per Rule 414.4, the cost can only be paid if the action can be completed, so you can't declare a move and skip it. There's no free "exhaust this unit" action, and note it wouldn't help anyway: Irelia, Fervent triggers on being chosen or readied, not on being exhausted.