It all depends if the listing is by itself or has variants!
If the listing does not have any variants, because Etsy doesn't allow listings to be adjusted with a stock level of 0 or below, Trunk will deactivate the listing and adjust it to a quantity of 1 (the lowest quantity allowed) to prevent it from being oversold. If the listing has been deactivated due to being out of stock, you'll see this:
Trunk will then re-activate it when it needs to go back in stock and then adjust it with the proper quantities. If you had originally deactivated the listing yourself, then the listing will not be re-activated since Trunk doesn't know why it was inactive in the first place.
If the listing does have variants, Trunk is able to keep the listing active and adjust just the variant as out of stock as long as one of the other variants is still in stock. However, if all the other variants are already out of stock, then Trunk will behave similarly as above and deactivate the listing.
Nothing will be deactivated if your Etsy listings actually become sold out themselves. Trunk will also renew your listings if they were originally sold out but now need to be put back into stock. Etsy charges may apply here but we're going to assume here that this is preferable over not having it in stock. Please let us know if you'd like finer control over this.