The relationship between the CoE and Russia is like a relationship in which there used to be some sincere hopes for positive feelings, if not sincere love then at least cooperation, mutual benefit, common purpose. Today, such positive feelings are more at the background.
In the context of the Council of Europe (CoE), there are two inter-connected yet slightly different levels. There is the political level (PACE, i.e. Parliamentary Assembly of the Council of Europe) and the legal level (the European Court of Human Rights, ECtHR). We can recall that Russia became member of the CoE as organization in 1996 and came under the jurisdiction of the ECtHR when the State Duma ratified the European Convention on Human Rights in 1998.
The problems that have emerged in the relations between Russia and the CoE have built up over the years. In earlier years, there have been judgments of the ECtHR that Moscow strongly disliked (e.g. Ilascu v Moldova and Russia; Kononov v Latvia in Grand Chamber). The ECtHR and the Russian Constitutional Court clashed directly in the Markin v Russia case, having opposing interpretations on what human rights meant in that concrete case. Of course, there were other really big cases like Yukos v Russia that were politically difficult and involved a lot of money.
Nevertheless, I would argue that the relations of Russia and the CoE deteriorated particularly with the annexation of Crimea and the war in Donbass since 2014. The voting rights of the Russian delegation in PACE were then suspended and the Russian delegates have not participated in the work of PACE ever since. On the legal level, the Russian Constitutional Court led by its Chairman Valery Zorkin decided in July 2015 that in the future, judgments of the ECtHR will only be implemented if they are constitutionally sound in Russia. This principle was later on made law by the State Duma. In a couple of cases (Anchugov & Gladkov; Yukos), the Constitutional Court has found that it is not possible to implement judgments delivered by Strasbourg because doing so would not be compatible with the Russian constitution. Yet it is not at all sure in terms of the CoE foundational rules whether the Russian Constitutional Court can really claim this kind of “veto right.” If all member states would claim this kind of veto, what would remain of the ECtHR and its authority to issue binding judgments?
Most importantly from political viewpoint, formally protesting the suspension of its delegation’s voting rights in the PACE, Russia has stopped paying its membership fees at the CoE since 2017. It is an interesting way of showing one’s discontent because Moscow knows that CoE must eventually exclude Russia – or at least suspend its membership – if it continues not paying the membership fee. Running the CoE and the ECtHR costs money and someone has to pay the bills. It seems that Moscow has hoped that this threat would make the CoE change its mind (and rules), and altogether become less “anti-Russian.”