Mare won't pick up her front feet

Take a soft cotton rope and loop it around here ankle.  Gently pull on it until she picks it up, even a little.  As soon as she does this, release the pressure.  Let her stand for a second and do this again, repeat.  Switch to the other leg.  This will get her used to picking it up without a fight and you not getting knocked in the head.
 
As you progress and start picking it up on your own, hold it for a minute, then PLACE it down.  Don't just let go and drop it as some horses do not like that sensation.  Then work to holding it up longer.  If she fights, pick it right back up.  As soon as she lets you keep it for the time you want be done. Reward her for doing as you ask.

Facing her hind quarters, reach down her leg to her fetlock, lean against her shoulder as hard as you need to to lift the weight up, and pinch that fetlock while pulling up her foot. Just hold it for a few seconds, praise the heck out of her, try it on the far foot. Add a little time each day.
 
I've had a lot of older horses who had the bluff in about their feet, lol. Stinkers. Some are always a pain, once they've gotten used to stomping it down. But it doesn't sound like your horse is doing that.

Hi,
Firstly, especially as she's good with her backs, I would be looking at the possibility of pain. If she's foundered, or even just laminitic, this can make them reticent to pick up a front foot, because it places even more weight on the other hurting foot. If you think this is likely, or possible, you can try standing her on some high density foam or such, so that she's more comfortable.

Re just training, I agree with the person who advised to do it gradually in baby steps, and if you feel it's safer, starting with a rope wrapped(not tied) around her fetlock can be helpful. In addition to well timed negative reinforcement(removal of the pressure - giving her foot back & quitting hassling), positive reinforcement(reward, Good Stuff) is also helpful. Saying 'Good!' as you give her a bit of carrot or such, while she's picking her foot up for you is incredibly motivating for them most of the time, if they're not afraid.

Reach under her belly and pick up the opposite foot - she can't lean on you that way!

I agree with everyone except when you set the foot down, don't just drop it or let her put it down. Hold it until you can put it down for her. This will let her know you are in charge and will help your farrier. You can also tap on it with your hand or a hoof pick.  She needs  to know when the farrier has her foot in his hands shes not to pull it back. So hold it up for 30 seconds or so and then put it down. I would use the rope around her foot first. Be safe