Does anyone have any thoughts on what effect slowing healing down might have on Spirit of the Century?  Specifically, what would happen if, instead of stress boxes clearing out after resting, the only way to clear out stress boxes was through medical treatment (i.e. first aid, medicine, etc.)?

I'm experimenting with making conflicts seem far more dangerous, even with minor hits being traded back and forth.  I've seen several propositions (including the recent blog post about Dresden), but I wanted some input about this (which is essentially taken straight from The Shadow of Yesterday).

If you wanted to increase the importance of healing and make conflicts seem more dangerous, the easiest way it seems to me would be to reduce the number of stress boxes.
I haven't run SOTC yet, but I've run a lot of FATE 1/2, and in that you had the following:
[][] Clipped
[][] Hurt
[][] Injured
[]  Taken Out.

In my games, I changed it to
[] Clipped
[] Hurt
[] Injured
[] Taken Out.

This improved the pacing of conflicts quite well. I think it's better than stopping boxes healing between conflicts, because that's extra book-keeping, and has an attrition factor - players get more reluctant to take risks later in an adventure when you're building to the climax, which is the exact opposite of what you usually want to happen.

With the Dresden thoughts, I'm now be tempted to use this in Spirit:

1-2: [] Clipped, no Consequence
3-4: [] Mild Consequence
5-6: [] Moderate Consequence
7+:  [] Severe Consequence
A 2nd Severe Consequence is a Taken Out result.

That's been one of my considerations, and I agree it's probably one of the simplest.  Just a note that I am trying to change the tenor of the game from action pulp to something a lot grittier; I'm potentially going to be using FATE for a Zombie apocalypse game.  Getting hurt is dangerous, and getting taken out means you become a zombie.

Notice the method I suggested above retains one element of SOTC: you can't get taken out on a single blow, unless you choose to be. The best result is a Severe consequence - it takes at least two blows.

Though a zombie game - if you want to model the attrition that appears in many of the movies - might be well served by your original idea of making sure wounds hang around.
Having said that, wounds do tend to be fatal - you might get a broekn bone or a bullet wound, but you don't wander around with multiple wounds. SOTC Stress isn't actually 'hits' - it's being temporarily disadvantaged, being fatigued, or losing morale or whatever. And zombie movie characters do go through a lot of stress, then come back for more. Then, occasionally, they take a serious wound - which is not unlike the SOTC consequence system could be used as is.
Maybe you should have two tracks: a temporary stress, and a long term stress (burnout). As they take burnout, that doens't heal anywhere near so easily, and taking a certain amount might cause them to wig out in the middle of a conflict.
You could have a refreshment system encouraging characters to bond together, which helps reduce burnout.
But burnout might be used to give them bonuses when acting selfishly (a bit like consequence aspects, really, or maybe it's a fixed bonus), so players are torn between bodning with each other, and then going all out for them selves when the stress gets too high.

I'm projecting what I'd like to see, but that sounds like a pretty good basis for a zombie movie.

Also, if SOTC consequences are used, someone has to get this one: "everyone things I'm about to turn into a zombie."

I think the discussion here's been good.  Take a look at www.dresdenfilresrpg.com for an idea of where I'm going with the Dresden Files stress track -- that may be more up your alley.  I don't think stress boxes should stay checked; consequences are where you want your persistence, since they go hand in hand with the aspect payout potential.