When cutting nails with dremel, can sparks be dangerous?
Cutting metal with an abrasive disk will always cause sparks, but they are so small that each spark (which is actually a very hot piece of abrasive or metal) doesn't contain much energy.  They typically can't burn your skin unless you concentrate them in one spot, and will not burn wood or other materials.

If you happen to have a gas can or oily rags or other highly flammable material, you should be careful but typical building materials shouldn't cause a concern.  Wear a hat to protect your hair (it can singe) and a face mask or good fitting safety glasses to protect your eyes.

The alternative would be to use something like a reciprocating saw with a metal cutting blade (basically a powered hack-saw).  This would probably put too much force on the nails and might cause them to loosen or even rise slightly.  It's also a heavier tool and would be harder to use.