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It is advisable to leave your router on all the time because if you turn it off you will not be able to access the internet via it and it could cause problems long term with the speed and stability of your line. These routers are designed to be on 24 hours a day, 7 days a week.
Regarding your original question, yes your Sky+HD box will continue to work if you do turn it off but you will not be able to download an on demand content.