Verb: to wash, to launder, to clean.
Pestä appears in our example in the imperative singular form, pese.
Let’s translate the whole sign, ”Älä pese astioita vessassa,” or ”Do not wash dishes in the bathroom.”
Älä is the negation verb in the imperative. It means ”Don’t!”
Astioita is partitive plural of astia, which means ”dish.” Here in plural, it means ”dishes” and is in the partitive because the phrase is in the negative and negative statements take the partitive. (If you were to make a positive statement, it would be ”pese astiat!” with astia taking the nominative plural instead of partitive.)
Vessassa is inesssive singular of vessa, meaning ”toilet” or ”bathroom.” So it means here ”in the bathroom.”
(Viemärit tukkeutuvat gives the reason why. It means ”drains will become clogged.”)