I'm interested in hearing answers from all rites of Christianity (especially the Syrians/Coptics; while I've seen a Byzantine monk remove his klobuk, I've never seen a Syrian/Coptic monk without his hood).
I won't speak for the Copts, but only for the Syrians.
When someone is ordained a priest, he receives a circular skullcap composed of seven triangular pieces stitched together. Each triangle represents an order of priesthood (the baptised, the "confessor", the chanter, the reader, the subdeacon, the deacon, and the priest). It symbolises the priest's sharing in all these orders, his passage through them, etc. It is technically a headcovering, but it is not treated as a headcovering, but rather as a necessary vestment.
Monks wear the monastic hood called eskimo
. It is the equivalent of the Greek monastic schema. IIRC, it is never taken off in the presence of other people, or even in private except for washing, shaving the head, etc. Monks who are also priests wear the skullcap under this hood. Why is the schema kept on even during prayer? I don't really know. It's a monastic custom and I'm sure had some sort of reason which derogated from the norm of uncovered heads.
Bishops wear, as part of their liturgical vestments, a garment called the masnaphtho
. It is sort of like an amice, but it can be worn on the head as a hood and retracted as needed. It is the equivalent of the Greek mitre. Even though the monastic hood and priestly skullcap stay on at all times, the bishop will slide this hood on and off depending on the part of the liturgy currently going on.