I can't follow most of the dialogue in Tatort, but the German subtitles I can understand almost completely.
Past episodes of Tatort are easy to find online, but the only ones I've found that have subtitles are those shown at www.ardmediathek.de.
Does anyone know how to get past Tatort episodes with subtitles? Alternatively, does anyone know where I can get just the subtitles for past Tatort episodes? (At the moment I'm interested only in subtitles German.)
Crazy as it may sound, you could buy the series on DVD, since they usually provide subtitles.
Now, as you can imagine, unofficial subtitles may be considered illegal. As discussed here (SE Meta Anime) and here (SE Law), “Subtitles are treated much like movie scripts by most video industries” and therefore StackExchange is not a good platform for this question to be asked.
That being said, and the legal question not fully answered, the subtitles for some episodes of Tatort may be found on this website.
Expanding my answer a little: You might also find what you are looking for here, here and here.
The official website let's you stream recent episodes with (German) subtitles. https://www.daserste.de/unterhaltung/krimi/tatort/videos
You can find some subtitles files (srt) at opensubtitles.org. Approximatively 50 files.
