Is it normal that a CRT heats when is off?

I'm no expert but it doesn't sound that unusual to me. For one there's a heater element in the gun. And I don't think the image is completely off when there's no signal but it should be reduced power. Think of it this way, they need to be warmed up to get good color saturation and then there's the speed of presenting the image (i.e. when you turn a CRT on it's not instant perfect image).
I could see the same set being warmer than another like it if they have different brightness settings (i.e. the black level being less black on one of the sets).

Some CRT-based TVs keep a small current flowing through the filament so that the TV starts up faster. Therefore, the TV's power supply and other components are active even if the TV is "off". For many CRT TVs, "off" is really more of a "Standby" mode.

I'm thinking he meant on but with no input signal ("...there's no image on its screen.") but after further review it looks like we can interpret it either way.