Despite several historical hypotheses of the Shroud that have been put forward involving Othon de la Roche, it is very unlikely that Othon had anything to do with the Shroud of Turin. The main reason for this conclusion is that the seed of all these hypotheses is the dissertation in favor of the authenticity of the Shroud of Besançon, written in 1714, contained in the manuscript 826 of the archives of the Besançon library. That is, all subsequent historical documents mentioning Othon de la Roche as possibly having owned the Shroud are based directly or indirectly to that dissertation. But that dissertation has no solid foundation to state that Othon de la Roche was involved with any shroud: the dissertation refers to documents that never mention that Othon received a shroud or owned any shroud. The book Le Saint Suaire de Besançon discusses these hypotheses and has a complete transcription of the manuscript 826 (in French).
Moreover, the chest still at the Ray-sur-Saône appears unlikely to have been used to bring any shroud back from Greece. …
And any hypothesis stating that the Shroud came to Lirey through Jeanne de Vergy (second wife of Geoffroy de Charny), who would have been a descendant of Othon de la Roche, is fraught with other major issues. For example, the receipt of Humbert de Villersexel, given in 1418 to the clerics of the collegiate church of Lirey, states clearly that the reliquary containing the Shroud had the coat of arms of de Charny, not of de Vergy. The son and the grand-daughter of Geoffroy de Charny also stated clearly that the Shroud was from Geoffroy de Charny, not from Jeanne de Vergy.