Suppose somebody asks the above question to you. Suppose that the reference of "that" has already been clarified, so perhaps they say something like "I heard that Muse's new album is rubbish. Do you know if that's true?"

I think we can re-phrase the sentence as so:

If that's true, do you know that that's true?

Now let us use the material definition of "if". We can answer in the following 4 ways:

Yes. It is true and I know it's true.
Yes. It is false and I know it is true.
Yes. It is false and I do not know that it is true.
No. It is true and I do not know that it's true.

Two of these answers entail Moore's paradox. The only consistent answers are:

Yes. It is true and I know it's true.
Yes. It is false and I do not know that it is true.

Only the first of these sounds like a reasonable answer to the question. If somebody asked us a question like "Do you know if that's true?" and we answered "Yes", it would be very queer if what we really meant was "It is false and I do not know that it is true." It's also impossible to answer the question except in the affirmative, which would suggest that the speaker asked a pointless question such that the most simple correct answer (Yes/No) could not inform them either way. But it isn't a pointless question and it can be answered in the negative quite sensibly.

So what are they really asking? Here are some interpretations:

1. Is that true?
2. Do you know that that is true?
3. Do you know the truth-value of that? And if so, what is its truth-value?

It's a tough one. If you answered "No", they might reasonably ask you "Is it false, or do you just not know?" If they asked this they would have to admit to having asked an ambiguous question in the first place, one that could be interpreted as either 1 or 2. "Yes" doesn't have the same problem because your assertion that it is true entails that you would assert your knowledge of its truth and your assertion that you know its truth entails that you would assert it is true. 3 is more often than not what they really intend, but we can't give them the benefit of having expressed 3 because they clearly haven't put enough work in to get 3. It's two questions and they only asked one and so should receive only one answer. Grice might argue that we really should interpret them as meaning 3, if that is most likely, but I'm not convinced that somebody wouldn't ask the question meaning only either 1 or 2.

As the only possible answer is "Yes", they should expect us to give that answer beforehand, thus rendering the question pointless. They could ask "Do you know if that's true? Is it true or do you not know?" but then they could have just asked the second of these questions on its own.

This question also reminds me of a few other kinds which can be interpreted differently so regularly that the asker of the question can't really ascertain the real answer from the one give, although they are not quite as sever as the one above. They are of the form:

Do you mind if X?
Are you sure you don't want to X?

For the first of these if you answer "Yes" to a question about whether you mind then you are giving your disapproval for the proposed event or fact. And if you answer "No" then you give your assent to it. But we equate "Yes" with assent and "No" with disapproval so readily that we (the questioner) might hesitate about the answer given wondering if our partner has understood us properly. If the conversation ended with their answer then I suspect both participants would leave unsure of what the other believed. The second example is similar. Because of the way the question is constructed, "Yes" indicates disapproval and "No" indicates assent (although it could mean indecision too). Given that none of us work on the assumption that anybody else is rational, and that they could interpret these questions several ways, we ought to just stop asking them.