I currently have a Sony 20-inch monitor and love everything about it
except for its loud fan. Does the EIZO CE240W make any noise (from a fan
or anything else)?
Also:
Does it generate a lot of heat (I have noticed this with Apple monitors.)
Thanks in advance for any comments.

There are no LCD displays that I'm aware of that have fans. If your Sony is an LCD, then that's the first I've ever heard of a fan in an LCD---or even in a older CRT.

Apple monitors hot? If you want to know what hot is, run a large CRT. That will turn a small room into an oven in no time flat. The heat output of the Apple LCDs (I have a 20") is no greater than any other LCD out there. Even if it is, it's not worth worrying about as the light-bulb illuminating your room puts out more.

LCDs are much more energy efficient than the technology they replaced (CRT).

And, this is a topic better suited in the Image Processing forum:
http://luminous-landscape.com/forum/index.php?showforum=5

Like the other poster, I've never heard of a monitor with a fan. I've gone thru lots of monitors at work over the years, and a few in my home system, and never had a monitor with a fan. As for Apple LCDs, I'm not sure where you get the idea they run hot. I have a 30" Cinema Display and I can run it all day and it stays cool as a cucumber.