As with the difference between direct quotes and indirect quotes, there is little difference in meaning between these two structures. The difference is in phrasing and nuance. The first structure is a straightforward question. The second structure is more like a quote, as if you asked someone that question directly.
Imagine these written as direct quotes:
Tell me what I should do?
Tell me, "What should I do?"
Do you know what you can do to fix the situation?
Do you know, "What can you do to fix the situation?"
Tell me where he goes on Tuesday evenings?
Tell me, "Where does he go on Tuesday evenings?"
These kind of inversions are common in English, and are often slightly more formal or literary (or at least affected). They therefore can be more dramatic in context.
"It's not the best solution," he sighed, "but tell me what else can we do in times such as these?"
Here the inversion is used as a rhetorical device to suggest the person speaking isn't really looking for an answer. He's just expressing his frustration. The other phrasing would better indicate that he wants advice:
"It's not the best solution," he sighed, "but tell me what else we can do in times such as these?"