Tube cage on or off during use

Can't say for sure. I listen to my VTL's with the cages off, figuring the more air circulating around the tubes the longer they'll last. That's probably just one of many misconceptions I have about electronics. My bet is that it has no effect on tube life. In my room nobody ever goes near them anyhow, so I'm not afraid of damage. As for sound, I imagine the louder the level the greater the chance of some cage rattling, but, again, I might have just made that up. Now for the real reason to listen without the cages: it looks cool.

My Fisher mono's sound better without the cages. Not sure if it is because of vibration/resonance, or, shielding.

If you want the convenience of leaving on your tube cage then suggest you absorb the microvibrations by placing a few Cardas Myrtle Wood Cuboids (myrtle wood vibration reducing/?tuning blocks) on the top of the cage (not directly over hot tubes) and beneath the bottom of the amp chassis. I got mine from Brian of Hellosimplymusic.com here on Audiogon.
Worked for me with one amp lacking removable feature. Unless the amp runs red hot the wood should stay in fine condition.
Hodie, Pearl Tube Coolers surround and help tubes to reach their maximim lifespan by wicking away heat more effectively than just sitting out in fresh air. Available from Parts Connexion here on Audiogon. But,it takes away from the joy of seeing glowing nude tubes.

I can't see how it would make a difference. I'd leave it off if you don't have kids or animals that might touch it.

Tube cages are really designed to keep someone or something from getting burned from touching the tubes. Some enhance the looks also. If safety is no concern then just go with what looks better to you.

Some tube cages do vibrate when placed close to the transducers, but most quality amps such as the TAD should not be a problem.

Thanks for the Pearl Coolers tip, Listener57. I'll have to weigh my appreciation of nudes against heat-wicking covers.