The technology in those picture tubes was very robust. The guns that fire electrons at the phosphors are almost indestructible, and the rest of the components are simple enough that you can find substitutes, should an identical replacement part not be available. Burn-in happens at the phosphor layer of the screen.

For LCD TVs the backlight will eventually die, it is after all for older models a fluorescent lamp, and if your TV is sufficiently old I'm sure sourcing the replacement part will become next to impossible. The newer ones with LED backlighting should last longer, if failures do not happen elsewhere.

OLEDs the panel itself degrades as it gets used, leading to screen burn-in. The screen itself might ultimately last longer than LCD TVs, but it might be left with a shadowy reminder of programs past watched.