Answer: Viburnums are potentially subject to some disease problems. Based on your description I think your viburnum shrub may have powdery mildew. This appears during certain weather conditions and affects plants such as lilac and certain viburnums. Usually this does not require treatment as it is primarily a cosmetic problem and will not kill the plant.
Some gardeners will spray preventively (as a dormant treatment in winter, or before the mildew appears, or at the very first slight sign of it) to protect their shrubs if it is a repeated problem every year. Since viburnums are sensitive to certain sprays (including sulfur which is most commonly recommended for powdery mildew) I would suggest you consult with your local county extension.
They should be able to give you a specific diagnosis (based on a foliage sample and/or photographs) and based on knowing that, make recommendations. If it is something that requires a chemical control they will have the most up to date recommendations on what to use and how/when is best to apply it.
In the meantime, pick up and dispose of any fallen foliage both now and in the fall, this will help limit sources of reinfection next year. Also, avoid wetting the foliage if you water the plant; if you water, try to do it early in the morning so the plant is dried off by nighttime. Good luck with your viburnum.
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