“Put it to Bed!” is a common way to say, perhaps not too politely, that it is time to leave that subject behind, to not worry about something anymore.
Sometimes, for example, someone may get annoyed that his friend keeps talking about what happened the other day. If he just won’t stop talking about it he may tell his friend, “Please, just put it to bed.” Interestingly, one would never usually say that you would “put it to bed” for any other English speaking situation – you would say, for example, “I will put the baby to bed” or “I will put her to bed” – but not that you would “put it to bed.”
There is a very similar expression which also has to do with sleeping and that is “Give it a rest!”. You can use “give it a rest” just the same as you would “put it to bed.”