I would not expect so, not unless it was common occurrence. It is not a lot of thymol in mouthwash and a single one time would be unlikely to be harmful. It is damage over a period of time. I would wonder if alcohol alone would do the same? I have never needed no resort to such a method know on wood I will not need to. I have seen a big retic bite (38 stitches) and can imagine what a feeding bite would be, the snake has got to release.
There are some whom believe mouthwash as a disinfectant which is not sound practice, phenols are know to be harmful. Two, diluted it is unlikely to do anything, and three even full strength it is targeted to a specific set of bacteria that need moist wet and really warm conditions, (human mouth) unlikely to be found inside an enclosure.