I have a related question. I always considered leaving the printer on was the way to go as it goes through a maintenance routine periodically. I have a canon 3650 , cheap unlike the print heads! There are various settings for power down . Googling the internet some say the print heads are 'capped' during power down so conflicting advice there. Any thoughts?
The printheads at shutdown / off-mode move to an specific place where a set of one or two caps (depending the printhead) press against them. It's rubber, and it's function is to avoid pressure differences that would allow ink to come out (cry, we call it). This also avoids any dust to come in contact with the heads.
Keeping the printer on for extended periods will produce the same as they are programmed to do it, even if it's still on. You can easily see this when work with a printer totally opened without the case.
Things are not crystal clear as diff models and brands = diff stuff. In the past some people left the printers on to avoid the usual power-on cleaning cycles and save ink, but due to the internal clocks, the some cleaning cycles will still take place. There is no perfect answer for this, as... the more expensive and specialized printers become, the more mainteinance they need and more ink they waste to avoid clogging.