I too doubt he was involved in the Hospital thing; but he was in the process of betraying Michael. He was setting up a hit on his territory for Barzini. The only thing that leads me to think that is speculation as Turnbull points out. The scene in the Don's study where the Michael is the Godfather and officially in power, where Tessio and Clemenza petition to break off from the family. Well on the most literal level; I think FFC and MP, used this to show the characters of Tessio and Clemenza, one stayed loyal but the other on seeked to betray Michael and I think it was Tessio's ultimate justification that under Michael's command he would only falter and so would his regime, so he thought he should do some "business" with Barzini.

However, that got me thinking. Clemenza, is very loyal and had been. Him petitioning to leave the family doesn't seem very typical of his character. Now this is speculation, but I think Tessio may have had a hand in convincing Clemenza that Michael wasnt' a great Don. Clemenza may have confessed to Tessio that he had doubts about Michael and Tessio may have convinced him that he was right. If this is the case it would imply that Tessio had plans to consolidate more power before the Godfather (vito) had died. If this is the case, it's slightly more likely that he had a hand in the hospital scene.