I've had mine on since the 8th December 2006, the day the Wii launched

Yes, I keep it on all the time.  Like TWRoO said, you won't keep getting messages if you stop using the programs that send messages.  If you stop playing Mario Kart, it will stop sending you messages after one or two months.  If you stop going to the Check Mii Out channel, it will stop sending you messages after just a couple of weeks.

I've had it on since launch.  No problems yet. 

Initially I was somewhat concerned with how hot it could get (they should keep the fan on low like when the PS3 has Remote Play enabled), but after a few months with no problems, I just stopped caring.  I figured Nintendo wouldn't include a feature that would break your Wii anyhow lol.

no I only keep it on when the system is on - this is for two reasons

1. I really don't need the things that wii connect 24 sends as soon as they send them out and

2. I heard that it's possible that leaving it on could burn out the gpu

i have always had mine on since the day i purchase it; the only time it was off was when i had to send my wii back to nintendo cuz the gears were getting ..noisy

i actually prefer it be on.. just because
plus i did a deal wit my gf to water my town in Animal Crossing so the majority of the time my wii is on... sometimes a week before it shut it off

I've always left it on.  Why not?

Yes, I leave it on.

I have it on now but I am thinking about turning it off.  One weird thing about my Wii is when I play NES games and hit the home button the buttons on the screen pixels start going out on the button.  I never have pixel problems like this while playing the game just when hitting home button during NES VC game.  Isn't the big of problem (maybe if pixels started going out while the game was playing then I would send it in).  Anyways, I think I'll turn it off since I find no use for it.  I agree with you d21lewis they could have used it to better effect. 

You will still get messages that your friends send you right (once you turn your Wii on)?

Just turned it off now.  I felt my Wii and it felt pretty damn hot.  Couldn't they release a firmware update that makes the fan run while WiiConnect24 is on?

hmm. So I had it on and felt side of Wii in back, it was warm to touch, pretty warm actually. I turned Wii off all the way and came back few hours later and it was cold. Massive difference. I'll just leave it off now. Need to change setting though so I don't have to hold power button on console.

Exactly what I concluded.  My GCN was cold (because it just sits there) while my Wii sitting right next to it was very warm.  Felt the back of it right after I turned it on and was blowing pretty hot air out.  I'll definitely leave it off for now on (unless they release a firmware update that lets the fan run while WiiConnect24 is on (if that is possible)).