Origen does not really have much of a role or influence in the activities of today's Church, apart from his being quoted every now and then insofar as he exemplifies Orthodox teaching and thought. Although he is not commemorated in the Coptic Synaxarium, he is nevertheless given various titles of respect (i.e. “Origen, the scholar”, or “the erudite Origen”) whenever his person is relevant to a Saint being commemorated (viz. St. Gregory the Wonder-Worker, and Sts. Barbara and Juliana).
As far as the erroneous doctrines usually attributed to him are concerned, the Fathers of our Church have clearly opposed them whenever they have been an issue of concern. So for example, the great Doctor of our Church, St. Severus of Antioch, wrote a brilliant patristically and scripturally supported refutation of the doctrine of Apokatastasis in response to a circulating rumour that he held to such a belief. In the appendix of his work Quintessence of Religious Doctrines, Mor Dionysius Geervaghese Vattasseril, a Syrian Orthodox Church Father of the 20th century, provides a list of heretics and a summary of their teachings; although he does not mention Origen, he mentions a certain “Stephen, the Son of Sundayli”, whose heresy, according to Mor Dionysius, was the belief that, “there is no eternal punishment as there is an end to every punishment. Therefore, the sinners will not be punished forever. Similarly, mercy will be shown even to the devils. In the end, everything will return to God.” The translator of this work, Prof. Matthew Oruvattithara, inserts his own comments in the section of this translated work pertaining to the end times. He states that the doctrine of Apokatastasis is “a heresy…which stood as a bar to Origen being canonised by the Church.”