I'm pretty confident in my DIY abilities. Although this will be my first foray into gas piping, I've done plumbing, electrical, framing, drywall, car repair, HVAC, etc.
Are there any specialty tools I'll need other than a large wrench for the connections on the tank?
You "need" two pipe wrenches for any black pipe work. I put it in quotes, because you can try to get by with one and some pliers or other nonsense, and if you're only taking apart a union, you will likely be OK with just one and slip-joint pliers. I'd calculate you're saving more than $300 by doing it yourself and buy the second $15 pipe wrench if you need to do any black pipe assembly and can't borrow one. Get gas compatible pipe dope as well.
Slip joint ("water pump") pliers are probably good enough for the water inlet/outlet connections. Worst case, use the pipe wrench and it'll just chew up the connections a bit but will work. Wrap with a rag to minimize the chewed up look (which will look like crap to a home inspector).
Copper piping needs emory cloth, flux, solder (lead-free!!!), a torch, and a wet rag. You can use white bread (no crusts) in the pipes to soak up a little bit of water.
If you need to do any work on the exhaust (maybe you can't find a perfect drop-in replacement), you might need a tub of refractory cement and you might need sheet metal screws to hold the exhaust together.
Consider adding ball valves to the water lines, at least the inlet, if they're not already present. Even if it's another cut and solder, I'd want to be able to shutoff the water to a leaking tank without killing the water to the whole house. In your situation, it will allow you to restore water to the rest of the house while you finish up the job.
To check for gas leaks, you can use soapy water. Others will use a match or your propane torch. (It will change noticeably in the presence of a natural gas leak.)
Take close-up pictures of the gas line T, drip leg, and gas inlet pipe. That's probably hard-piped and you might need to adjust the leg (with different sized nipples) if the height on the new tank is different. Take a shot from the floor to the gas inlet with a tape measure.
Take close-up pictures of the exhaust outlet and sheet metal chimney. Measure from the floor to top of existing tank.
Try to match those measurements or at least have a plan for deviations.
With your DIY experience, I think this is well-within your abilities. I measure DIY projects in "expected home center trips". This one is a "two trip" job, but it's not particularly difficult. :-D