Yes, priests are required to recite the Liturgy of the Hours. A diocesan priest would typically be bound to the Office of Readings, Morning Prayer, one Daytime Prayer, Evening Prayer and Night Prayer. I would imagine bishops have a similar obligation.
Permanent deacons are bound according to the prescript of his bishop and/or the country’s episcopal conference. I understand Morning Prayer and Evening Prayer are what are typically required of them.
Religious priests are bound by their Constitutions, and for those in the monastic life, they are bound not only to pray all Hours (including all three Daytime hours), but also to chant at least some of them in choir.
And although the phrase is no longer used, because the obligation binds upon them by law, having received a *mandate *from the Church when they were ordained, they are required to recite them under pain of sin, excused, as always, by a serious reason. To intentionally omit, without serious reason, the hinge hours of Morning Prayer or Evening Prayer can even be mortally sinful.