Is it true that some organs (such as the liver or the intestines) are less active at certain times of the day, regardless of whether the individual is awake or not?
No, this is not true. The only "organ" (if you will) that needs to "sleep" is the brain, and even then it is still quite active. The body needs to rest.
When blood is needed in a prioritized manner in certain areas of the body - for example, during a marathon the muscles and the heart need more oxygen - mechanisms are in place to shunt some of the blood away from organs that need it less (the stomach and intestines, for example) to those that need it more. But that doesn't mean the GI tract is sleeping, or even resting.
There is no organ that "sleeps". The brain as a whole is highly active during sleep, even secreting certain hormones predominantly during that time.