NYCFarmboy:don't hit me ...
But that's the only way to tune an organist!
myorgan:Why would you want an out-of-tune instrument? While I realize that's what gives orchestral strings their "warm" sound in an orchestra, what makes it so desirable in an organ?
It is basically the same thing; a slight mistuning of foundational tones causes a sort of heterodyning of the tone, which generally warms up the sound in orchestra or organ. If every pipe in the organ was absolutely spot-on, it would sound rather artificial. The main thing is that
slight tuning anomolies will tend to warm up the sound; gross ones will be...gross!

This is most true for foundational tones, less true for higher pitches.
myorgan:Isn't that what celestes are for?
No, celestes have a different purpose. They are typically soft string stops which make an ethereal quality. When louder or more foundational stops go too far out of tune it is much less pleasant.
myorgan:Could it have become acceptable because in centuries gone past the organs couldn't be kept in tune due to older heating techniques, so the tuners convinced churches that it was more desirable to have an out-of-tune organ?
Not exactly. Looking at Europe, you see that the larger stone churches tend to be rather cool year-round; those old cone-tuned organs tended to stay in tune pretty well with only periodic touch-ups of the reeds. This situation prevailed for many centuries. The relatively recent appearance of central heating brought more pronounced tuning problems.
myorgan:Kinda like how people centuries ago settled on the equal-temperament tuning scheme because they were convinced it was a better sound.
Don't know if it was a better sound, but equal temperament at least allowed people to play in all keys. If you are accustomed to unequal temperament you will find that some keys sound much better, and some keys sound much worse.
The higher the pitch and the louder the sound, the less I tolerate 'out-of-tuneness'. So I typically will tune all of the reeds very carefully, then I will go through the mixtures. Mixtures are a real pain to tune, so I usually get them close but not absolutely perfect. Most music played at speed will mask the effects of a mixture note being slightly out of tune.
This underscores the importance of having a notebook for the organ technician. If you hear something that is really out, you should write it down so it will be fixed at the next tuning (if not sooner, if it is really an emergency). 
Soubasse32