Please let me know what you think
is correct. Even if do were to be included, the correct form would be
Please let me know what you do think
but this should only be used for emphasis (for example, to contrast with don't think).
"Do you think they shouldn't attend that school? What if they would become great musicians?"
This example sounds awkward and mildly ambiguous to me. The ambiguity arises from the fact that would can have the meaning "strongly desire [to]," as in the movie title The Man Who Would Be King. So one reading of the second sentence above is:
What if they strongly desire to become great musicians?
But I suspect that the speaker has a different meaning in mind. Unfortunately that meaning is somewhat obscured by the omission from the second sentence of the intended condition under which "they" would become great musicians in the what-if scenario—namely, that they attend the school. Adding that condition to the second sentence is easy:
What if by doing so they would become great musicians?
or:
What if they would become great musicians as a result [of attending it]?
By introducing the condition into the what-if sentence, we avoid any possibility that readers might read the would in that sentence as meaning "strongly desire [to]." The use of "would become" in place of "became" is somewhat colloquial, but in an informal setting it seems reasonable enough.
"Do you think they shouldn't attend that school? What if they became great musicians?"
Because would has vanished from the scene, the second sentence in this version of the example doesn't have a built-in ambiguity. But the example still works better (I think) if we add the implied condition to it:
What if by doing so they became great musicians?
or:
What if they became great musicians as a result [of attending it]?
Is there any sentence where what if + would is grammatically correct?
It's not hard to imagine snippets of dialogue in which using "what if" + "would" is the most coherent and natural-sounding way to handle a hypothetical scenario. For example:
Person A: I bet you'd eat a whole coconut cream pie right now if you could.
Person B: What if I would?
Best Answer
think fit (alternatively see fit) is a set phrase meaning *to regard as proper or appropriate (Collins 1), or desirable (2).
Set phrases can seem to violate a grammar rule, but once you recognize them as such, you can try substituting their definition to see how they work.