I am one of Jehovah's Witnesses.
Jehovah's Witnesses have not published any comment on a particular religious organization being behind the court cases. They are much more concerned about the outcome of the cases happening at present. Religious persecution is not new, and is rather seen as a sign of the Great Tribulation.
There have been many interviews with non-JW experts at court cases over the years, and these experts indicate that the source of claims against Jehovah's Witnesses most often comes from people with a personal agenda against Jehovah's Witnesses.
Since the majority of people in Russia are Russian Orthodox, it's more likely that these individuals come from that religion, but I don't know of any evidence of the Russian Orthodox Church itself encouraging these specific court actions. It is more of a debate of how much control the Church has over the Russian Federation in general.
Here are quotes from experts which mentioned the Russian Orthodox Church in their interviews:
“For 25 years, there have been different legal, political, and intellectual conditions in Russia and Ukraine. I am told, though, that impartial theological expertise has virtually ceased to exist in Russia. The experts—as a rule, scholars—do not have the same status and are not protected by any special legislation. It is biased people who are denomination-oriented, usually toward the Orthodox religion, who are drawn to expert examination. Being Orthodox, they cannot analyze any other way of worship in a neutral and unbiased manner. Perceiving any other religion as not true, they, in their ‘righteous indignation for the purity of their faith,’ emotionally and aggressively brand all others as extremists.”
—Dr. Liudmyla Fylypovych, professor, head, History of Religions and Practical Religious Studies Department, Philosophy Institute of the National Academy of Sciences; vice president, Ukrainian Association of Researchers of Religion, Ukraine
“Elements within the Russian Orthodox Church connive with the forces of order to promote their own interests and to suppress any perceived competition.”
—Dr. Jim Beckford, fellow of the British Academy
“Basically, the law is overbroad. The language of the law can be used by law enforcement to arrest or threaten individuals with religious views that are unpopular or the government simply doesn’t like. Common examples include arrests of Jehovah’s Witnesses, other minority groups, and even atheists. The law has essentially been used to protect the Orthodox viewpoints that are supported by the government, and to punish viewpoints that are perceived as alternative to or threatening to this Orthodoxy.”
—Ms. Melissa Hooper, lawyer, director, International Law Scholarship Project/Pillar Project, Human Rights First; formerly regional director for American Bar Association Rule of Law Initiative in Moscow, United States
“A true Russian, if Christian, is a member of the Russian Orthodox Church. Russians belonging to the ‘wrong’ religious organization are isolated and excluded from society. Therefore, the civil rights of Jehovah’s Witnesses are being grossly violated.”
—Dr. Gerhard Besier, professor emeritus, European studies, Technische Universität Dresden; lecturer, Stanford University; director, Sigmund Neumann Institute for the Research on Freedom and Democracy, Germany
All of these quotes were supplementary to a 3-Part Article published by Jehovah's Witnesses: