This must not be confused with “Tell me what that says”, which is a command.
The sentence for which this thread was born starts out like a command – “Tell me” – before the interrogative portion is launched. This is a fine point, but to hammer home the obviousness of the need for a question mark, I’d include a comma:
“Tell me, what does that say?”
To recap:
Interrogative: “Tell me(,) what does that say?”
Command: Tell me what that says.
The word “what” does not necessarily connote a question.