What type of saw should I use to raise the legs of my hearth woodwork?
You'll either use a flexible, fine-toothed handsaw or what's called a "jamb saw"--a circular saw set up to cut very close to the floor. Which you choose is your prerogative. 

If you use a hand saw, which may be adequate for just two cuts, lay a piece of sheet metal on the floor (and height spacer, if needed) to protect it, and use your hand, protected by heavy gloves, to keep the saw tight to the floor. 

With either saw the biggest risk is damage to your woodwork through splintering. Always direct the cut into the workpiece to prevent blowout.