2005 3 floor wire harness
 

So, my new puppy decided to chew through the wire harness that attaches to the driver seat. Managed to splice the wires together, but the air bag light will not go out. Suggestions (besides a muzzle for the dog)?

did you just twist the wires, or solder them? I know one of the airbag wires is a shunt (low resistance) and if you just twisted the wires it may now look 'out of spec'.
Also, since I assume the car was running, it may have logged a fault for the airbag which won't turn off on its own. Could need the computer code cleared, or maybe just a good ole battery reset would clear the light - assuming the wiring is all back to normal.

Problem is that the dog didn't just cut the wires, she chewed a piece out, so I had to splice a piece of copper wire inline. I soldered it in, but I'm not sure if the resistance is the same.

as long as its soldered you should be ok.
Disconnect the battery and see if that helps, or try the fuses for the air bag system (F65 & F71). The shunts I was talking about I looked up and they are in the 2 ohm range, way above a wire splice... which makes sense since there are a number of connectors in the system that add resistance too.

If cutting power (battery / fuse) for a minute doesn't turn the light off you may need the dealer to clear the code for you, I don't think the normal scan tools would see this code but it's worth a try if you have a friend with a scanner.