On the heels of the Supreme Court's decision in Obergefell v. Hodges, the Ohio Supreme Court's Board of Professional Conduct has issued Advisory Opinion 2015-1. The Syllabus of this Opinion states that:
"A judge who exercises the authority to perform civil marriages may not refuse to perform same-sex marriages while continuing to perform opposite-sex marriages. A judge may not decline to perform all marriages in order to avoid marrying same-sex couples based on his or her personal, moral, or religious beliefs."
The full text of the Opinion can be found at this link, and the Supreme Court's press release can be found at this link.