It helps me to translate as literally as possible and it can be kind of fun, just to see how very different syntax can be between languages: "I no know what hours they are."
"Don't" is just a contraction (meaning short for) "do not". When speaking to people, the use of the longer "do not" explicitly states the impossibility.
Regarding this sentence: I don't know what time it is, but I might check my watch (or cellphone these days) I do not know what time it is, and I have no means of checking it
Someone gave you a down vote. Probably because there really is no difference. I always say "I don't know what time it is" because it's easier to say. If I would say "I do not know what time it is", it would sound more emphatic, particularly if I put the emphasis on "not", which one can't do with the contraction. But as different people use language differently, this is just my opinion and may not apply to others.
If you say ‘I do not know what time is.’, then you are expressing your ignorance in a philosophical discussion about the nature of time. What is time? How is the future different from the past? Why are we inexorably drawn into the future, and can this be changed? The top Google hit for ‘What is time’ (from where I am) is http://www.timephysics.com/ which discusses these big-picture issues.
If you say ‘I do not know what the time is.’ or ‘I do not know what time it is.’, then you simply don't know the hour of the day. This is not a deep philosophical question, but just an every-day matter. Is it 4:00 yet, or not? The top Google hit for ‘What is the time’ is http://www.timeanddate.com/worldclock/, and the top hit for ‘What time is it’ is http://time.is/. Both of these are just clocks.
(If you say ‘I do not know what the time it is.’, then you're not saying anything correctly; that's just bad grammar.)