ALWAYS unplug the lamp and allow it to cool before moving it. That's basic safety.
Heat plates *can* be good -- but they have their limitations. For one thing, many of them are not rated to be used with ambient temperatures below 50F (look at the manual). For another, they're not capable of warming chicks that have been seriously chilled in shipping -- a fact a number of people discovered by having entire shipments die.
A properly-set-up, well-secured heat lamp presents little risk of fire, but they are not safe when mishandled.
