CRTs will get dimmer as time goes on for a variety of reasons. How fast depends on how high you set the brightness, how long they are on every day, ambient temperature etc. They can certainly last decades but they won't be as bright as they were new.
LCDs could last indefinitely (or at least as long as any other solid state device) except for the backlights. These may gradually get dimmer and eventually die. Cold Cathode Flourescent Lights (among the most energy efficient backlights) have typical lifespan ratings of 15000-50000 hours (5-17 years at 8 hours a day) while LED backlights could easily last 100,000 hours (34 years at 8 hours a day). The CCFL lamps themselves are not particularly expensive but they are generally soldered in place so replacing them might be labor intensive.
This could result in a supply of refurbished LCD monitors in 5-10 years as the cost to replace the CCFL backlight will probably always be less than the cost of a new LCD assuming apropriately sized tubes are still available.
Max L.