Many Catholic Churches offer Mass every day, sometimes more than once per day. Which other churches do this? (I'm particularly wondering about the Eastern Orthodox, Oriental Orthodox and Church of the East)
The short answer is yes, they do.
In the Eastern Orthodox Church, what is called "Mass" is usually referred to as "Divine Liturgy" or "Liturgy", although I have seen a Prayer Book from the Antiochian jurisdiction also use the term "Mass".
There is a strict rule within the Eastern Orthodox Church, however, that Divine Liturgy cannot be served at the same altar more than once per day nor by the same priest(s) more than once per day. Also, the Sunday Liturgy is only served on Sunday - there is no Saturday evening Liturgy that takes the place of Sunday morning (usually on Saturday evening a Vespers service takes place, preceded or followed by Confession). Large cathedrals and monasteries may have multiple chapels that allow more than one Liturgy to be served on-site each day, but by different priests at different altars. (I seem to recall that there is some ancient Church canon governing this practice, but I cannot find it).
There is nothing prohibiting the Divine Liturgy being served every day, except during those days of the year when there is no Divine Liturgy (e.g. Great and Holy Friday - "Good Friday" in the West). I think that within most jurisdictions, the bishop tries to have individual parishes schedule at least one Divine Liturgy during the day on different days of the week, so that there is one Liturgy available each day. Within my parish - which is very small - we have Divine Liturgy three times per week during the normal year, and sometimes every day during the Lenten and Pascal seasons, depending on the calendar. In addition we have usually have Vespers services on Wednesday and Saturday evenings and on the eves of major feast days. Many Orthodox monasteries celebrate the Divine Liturgy every day, again depending on the Church calendar.
Within the Eastern Orthodox Church we celebrate four types of Liturgies:
The Liturgy of St. James, which is very ancient and very long and is now celebrated only during the actual Feast of St. James
The Liturgy of St. Basil, which is quite a bit shorter and is celebrated during special seasons (e.g. Great Lent on Sundays).
The Liturgy of St. John Chrysostom, which is the "normal" Liturgy
The Presanctified Liturgy of St. Gregory Diologist (Pope St. Gregory the Great), which is said weekdays during Lent.
I am not completely sure, but I expect that you will find that most or all of what I describe above also applies to Eastern Rite Catholics under the jurisdiction of the Pope.
[Note: It is not relevant to your question, but I am a convert from Roman Catholicism to Eastern Orthodoxy, fairly steeped in the western monastic tradition (Benedictine)]