While I'm not familiar with the other Oriental Orthodox jurisdictions, in the Coptic Church, there is some discretion made in regards to mental fitness. HG. Bishop Youseff's full answer can be found here: http://www.suscopts.org/q&a/index.php?qid=1541&catid=689
but here's a short quote:
Suicide is seriously denounced in the church. In these horrific situations, the grief stricken family naturally needs special comfort through intense prayer and closure, which only the church can provide through the traditional funeral and burial services. The church uses wisdom, discretion, and compassion to first determine the cause of death. If it is known and documented that the person who committed suicide was mentally ill, the church will pray over the person in the common tradition of the prayers on the reposed. If in fact, it is determined that it was indeed suicide, the church may refuse the customary prayers for the reposed.