In some people's sink area they keep a glass or metal cup to wash their mouth out after they brush their teeth. Must one Tovel that cup even though they don't swallow the water/liquid? Plus, even if one does swallow, it is accidental.
The OU has the following on their website.
Secondly, the determination of tevilah depends on the owner's designated use for the object: a utensil purchased for non-food purposes and occasionally used to hold food (such as a screwdriver which might be used in a pinch in the absence of a fork) does not require tevilah (Aruch Hashulchan, Y.D. 120:40)
I would say a mouthwash cup is being used for non food purposes and does not need to be Toiveled.