No, there isn't.
Understand the rule for the use of say versus tell: we say something; we tell someone something.
So what is the story with: She told the truth; he is telling the truth. Is there ever an instance when usage would be: say the truth?
No, there isn't.
Hmm. He told the story interestingly - there doesn't have to be an indirect object. Perhaps you have a dodgy 'rule'. It's rather Ptolemaic to start with a bad rule and then think up reasons for 'exceptions' to it.
This is an interesting area. I believe that a hundred years ago or more it was possible to 'say truth' (no 'the'). If you think about it, 'the truth' is not what is being told; the speaker is being truthful. But if a child is adopted and finds out as an adult, he might say 'For 25 years everyone's been lying to me, and only now have I found at the truth' - he doesn't mean the truth that there are 5280 feet in a mile!
So the concept of 'the truth' (and telling it) seems like a fairly recent one. The 'telling' might be analogous to obscure (and now archaic) uses of 'tell', like 'telling one's beads' (saying the rosary). But I am not justifying an exception to a rule. I am saying that rather than think about a world of rules and exceptions it might be more fruitful to do a bit of digging in historical dictionaries.
You could them that 'tell the truth' is a fixed expression.
Why is the title of your thread "MJW"? It has nothing to do with the contents of your post. Remember that the title of your posts should include all or some of the words you are querying. A good title would have been "She told the truth" or similar.
Remember - if you don't use correct capitalisation, punctuation and spacing, anything you write will be incorrect.