It's rare that a good defensive catcher is moved to another position later in his career. The only one I can think of is Johnny Bench. He was moved to 1B at the end of April in his age 33 season, and played mostly 3B for the next two years, after which he retired at 35. Bench also happened to fall off dramatically on offense at the same time: 3.3 oWAR in his last year at catcher, 3.8 oWAR total in those last three years as a part-time player. He was not very good defensively at 1B or 3B.
The next best defensive catcher I could find who was moved was Yogi Berra. Yogi apparently was not as good defensively as JT: 9.2 dWAR in 14 years before he was moved to LF (JT has 8.9 dWAR in 9+ years, and has had three years with higher dWAR than Yogi's best year). Yogi was older than JT when he moved; in his age 35 year he played a third of his games in the outfield, and 85% of his games in the outfield the following year. Yogi actually had more games at catcher than outfield after that, but he was primarily a backup to Elston Howard.
The other catchers I can think of who moved to other positions (or DH) were not very good defensively. They are (with age they first played fewer games at catcher than at another position): Mike Piazza (35), Joe Mauer (31), Ted Simmons (34), Victor Martinez (32), Mickey Tettleton (32). All except Mauer had negative dWAR for the rest of their careers. Martinez, Tettleton, and Piazza had some good offensive years, not comparable to their mid-20s, of course, but still good enough to stay in the lineup. Mauer was just OK, and Simmons declined. All of them were better than JT offensively prior to their position switches.
My conclusions are that the catchers who change positions were usually not good defensively at their new positions, and they had varying degrees of age-appropriate decline offensively. Not a ringing endorsement of the idea of moving JT to another position.
(All WAR from baseball-reference)