- Do you know what time it is? right
- Do you think what time it is? wrong
- What time do you think it is? right
- What time do you know it is? wrong
The posing of the two questions together is a little confusing because the issues are distinct, but here's what I think:
Short answer: The first one is ungrammatical because 'what time it is" cannot be the complement of 'think'. The second is grammatical but violates usual pragmatic constraints.
Long answer: The first step is to simplify the issue by getting rid of the distractions posed by the interrogative nature of the main clauses. The first pair, as declarative main clauses, becomes:
- I know what time it is.
- *I think what time it is.
- I know it is this time.
- I think it is this time.
Bare declarative content clauses look like normal main clauses. It is this time is a good example. (Expanded declarative content clauses are exactly the same, except that they are prefaced by subordinator that. So, the expanded form of our example is that it is time.) On the other hand, interrogative content clauses, like questions, begin with one of the interrogative words: who, where, how, etc (note: whether is not one of these). Unlike many questions, however, open interrogative content clauses do not undergo subject-auxiliary inversion. So, the questions is what time is it, while the open interrogative content clause is what time it is.
Now back to our two verbs. Although, know allows both types of content clauses as compliment, think does not allow open interrogative content clauses. Rather, preceding an open interrogative, think requires about. This by itself deals with the first pair. It tells us that *I think what time it is and *Do you think what time it is are ungrammatical because the verb is being matched with the wrong type of complement. They should be I think about what time it is or Do you think about what time it is?
Now we return to the second pair. Despite the original claim that what time do you know it is is wrong, it is not ungrammatical (there's a difference). As we can see when we take it out of question form, the verb know clearly allows a bare declarative content clause as its complement. Still, it does seem odd.
I think we have a problem here with pragmatics, not grammar. When we ask somebody a question, we're normally asking about facts rather than a person's knowledge, so we ask, "what time is it" (generally, not just in your head). If you believe that a person's personal idea about the time differs from the facts, then it doesn't qualify as knowledge, so you would ask about thought, opinion, belief, etc. This applies to anything, not just time.
The question isn't a complete write off though. Imagine it's two o'clock by your watch, but your friend, says, "I know it is not 2:00." She claims to know that it is a different time while, unhelpfully, not telling you which time it is. In such a situation, you might be able to employ, "OK, smart ass, what time do you know it IS?"