How long can a stacked carrot cake be stored

There are two main considerations here -- temperature, and icing.

For most stacked cakes, you typically don't want to store them in the fridge once they're iced, because the chill can cause condensation on the cake once it's removed from the fridge, which can adversely affect the job that you did icing it.

If your room temperature is relatively cool (ie, it's winter and you don't blast the heat, or it's summer and you're running the air conditioning), with most cakes, you're fine leaving them at room temperature for up to three days, with or without icing, although if without, you'll want it covered or wrapped to prevent it drying out. (personally, I always bake my cakes the day before I stack and frost them, so that they've had a chance to thoroughly cool before splitting the layers apart).

My real concern is that carrot cakes often get a cream cheese frosting, and I would want to keep that refrigerated. If you're not using cream cheese, then you likely don't have to worry, but if you are, I would bake the cake two days in advance, let it cool to room temp overnight, stack it (or at least, each tier), then put it in the fridge, and ice it the day of the event.

If that would be too tight of a time frame on the day of the event, I'd ice it a day or two before, but do whatever I could to prevent condensation. If it's the sort of icing that crusts up a little bit, you might be able to let it sit out 'til it's set up well, then wrap it in plastic wrap / cling film, refrigerate it, then pull it out the morning of, let it get to room temp, then unwrap to do the final stack. If it doesn't crust up, I'd look into getting appropriately sized cake boxes or other containers for when you store it in the fridge and let it come back up to temperature.