I have several (around 6 or so) bricks on the exterior of my home that are spalling. How big of a concern should this be? I am planning on doing some tuckpointing soon, would you expect this to stop the spalling? Do I need to replace the damaged bricks?

A lot depends on the kind of bricks you have...bricks can range from sponge-like to almost completely impervious, and anywhere in between. The spalling is more likely due to the porosity of the brick than anything having to do with the mortar or joints.

If you can match the brick, chisel out the spalled ones and replace them before you tuck-point.

You can also use a siloxane based water repellent for bricks and masonry surfaces. This will keep water out of the brick to begin with and prevent the spalling.
brick sealer

Some times the builder uses the wrong type of brick for the application intended. Some bricks are used as pavers, some as siding and others as structural. Each like Richard said have diffrent hardness. Replace the bricks if they are bad enough visually. You will have a hard time matching them, you may be better off either not changing them at all if there are only a few to perhaps creating a pattern effect with several new bricks so it looks like the wall was built that way.
As far as sealaing the bricks. I strongly disagree with skootch on that one. Brick sealers are not suggested. They really do not work as they claim. A clear liquid silicone cannot bridge the gaps on brick surfaces. Bricks that do get some sort of moisture behind them need to breath to let it out. If you seal them they will crumble even faster. More information is available at www.bia.org
Your best bet is to determine what caused the failure and fix it. Not simply paint over the issue.

...with one exception. There are solvent-type acrylic coatings that can withstand a blast froma fire hose, yet still allow moisture to pass through the coating and out of the wall. They aren't cheap, but they work and work well.