I don't think it is possible to output a signal from the Display-out ports or in fact any other port when the PC is in sleep.
As an alternative solution, I can only think of three things, none of which are simple enough to be worth the investment, by that I mean it's easier to just buy a new TV. But here they are:
An IR blaster that sends the same signals as your TV remote, thus keeping the TV alive.
Many TVs have secret settings, which can be accessed by pressing a combination of buttons, but this is TV specific, so your best bet is to search the model of your TV in google and type something like, hidden settings, secret settings or similar behind it.
Maybe an active display switch that could automatically switch between inputs, but you would still have to connect a video source to the other input, something like a raspberry pi that would display a black image.
But there is a problem with solution number 1 and 3, the back-light would stay on which wastes electricity, so it isn't really economical, at that point as I already said, you are just better of buying a new TV/Monitor...