Remove Masonry Nails from Concrete floor

I'm a little lost. First, what are you using to attach the furring strips to the concrete? I'm hoping some construction adhesive and a few concrete screws/anchors to bear the weight You could use the same thing to attach the Hardee-board directly to the concrete (with a layer of moisture barrier separating them to keep moisture from the concrete seeping into the siding). I'm also not sure why you're telling us what the garage door is framed with if these 2x6s aren't a substantial part of the anchor for the furring strips.

Anyway, assuming it cannot be done any other way, I'd say that using 1" nails where 1-1/4" is recommended is acceptable; maybe use a few more along each furring strip to compensate for the decreased "bite" of the shorter length of each nail. Understand that a pneumatic nailer will drive the nails down beneath the surface of the Hardee-board, so even with 1/16" clearance that 1" nail may contact the concrete. The only additional recommendation I would give is to nail in at a slight downward angle where possible, as if you were hanging a picture. The angle will not only give you a little extra depth to drive the nail, but the angle will increase the leverage the nail has to hold the siding, like using a nail to hang a picture.

If you have acess to Dremel or other rotary tool you can use a cutoff /abrasive wheel attachment to cut them off. They may have been inserted by a power tool that fires them in with a gunpowder charge. If this is the case prying them out will be difficult if not impossible. You could try a Sawzall but these type of fasteners are very hard and you will go through a lot of blades. Some type of cut off wheel or grinder is the way to go. You may forgo the expense of buying one by renting one. If you opt for the grinder you will be able to grind away a little of the concrete so the nail will be below grade. You can then patch the low spots to level the finished surface. Be aware that any type of cutting wheel you use will throw sparks and if you grind the concrete it will generate losts of dust so use appropriate safety precautions.