Find a body that will stick inside the head. Depending on the differences in time between when certain bodies and heads were made, some can be really loose together and some can be really tight. Find a tight one - like, really tight, so that turning the head is difficult. Then try and pull it off slowly, pushing downwards on the face as you do. A word of caution: this method borders on nonsense, and probably won't work.

Then there's water. Is there any merit in the running these under a tap (upside down) and then leaving them out for a while (still upside down) before trying the method I stated above?

I am currently having similar issues. Helmets are stuck in my two LEGO bin pieces, and won't come out. I didn't stick them there - it was just from rambling around in my box and getting stuck by coincidence. Nothing I've tried has got them out, so I'm perhaps not the greatest authority on such subjects...:shifty:  I hope you manage to get them out, though!

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You might try putting the head onto a thick neck and just slowly twist and wiggle the helmet from front to back as best you can.  That usually gives enough leverage to be able to pull it off the head.  If you can kind of hold your finger on the front of the face while you do this it can help hold the head all the way on the neck while the helmet (hopefully) can wiggle off.  Getting the sticky stuff out of the helmet might be tough but at least you'll have the heads out.

EDIT: I guess I said basically what jampot suggested but with the wiggling.  Wiggling is good.  This way has helped me get helmets off before.

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I meant using a pair of really thin pliers to reach inside the head itself, which would prevent the neck from getting too damaged.  You could try also wrapping the nose of the pliers in something like duct tape, which would prevent scratches--though it might affect the grip.

Also, try using some of the same kneadable eraser on the neck of a torso, that way the head would stick to it much more.  It might be difficult to get off the torso afterwards, but it should be a lot easier than getting them out of the helmets.

This is something I've never tried and may not work at all- but some mini figure heads have small holes inside the stud on top, perhaps you could bend a wire such that it could be poked up inside the head and hooked through the hole, then used to pull the head out that way? Just a wild suggestion if all else fails.

Insert a pair of thin pliers into the head and open them, then pull the head out while keeping the pliers open.

Insert something, like a screwdriver into the head. Pull while keeping enough pressure on the inside of the head.

These methods will damage the inside of the head.

He doesn't want to damage them in any way.

Sometimes you can open a tight jar lid by putting it under some hot water, which makes the metal expand. Could something like that work with plastic as well? Or maybe it could at least help dissolve the "sticky stuff"..

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I left the heads in a cup of water in an attempt to soak the stick and tried that goblet idea with my tightest goblet.
Sophie's head is free!

Thacker and Wolfram are still stuck, though.  The goblet won't be enough as it is.  Sophie was the only one where I applied the stick myself, Wilson did it with the other two.  Wilson uses a lot.

Would butter work?