info-beamer hosted supports CEC for HDMI connected screens.
Powering off/on the attached screen
The power saver package can be added to any setup. Just follow the instructions in the linked page. It allows you the turn the attached screen on/off based on a schedule. This package uses the info-beamer OS API in the background. The Pi will first stop the info-beamer visualization process. It will then stop generating output over HDMI. For some older screens this might be enough to put them into sleep mode. Finally it sends the CEC standby command.
Turning on works in reverse: First a CEC poweron command is send using CEC. Then the video output is restarted and finally the info-beamer process is turned back on.
When you add this package to an existing setup, it can take up to a minute before the screen turns off, so be patient. Similarly, if you make changes to a schedule, the screen might wake on but is then put to sleep again.
CEC for input
info-beamer OS supports CEC input. So you can trigger commands from your remote if the visual code supports this. Here is a tiny CEC demo that shows this in action.