Replacing toilet shut-off valve and need to turn off water; Need to turn gas water heater to pilot?

Do I need to set the water heater to pilot, in order to avoid the dangerous scenario outlined above? Doing anything to our water heater makes me a bit nervous.

NO, In the amount of time that it will take you change the valve there will be no issues with it set to the current temperature.

Once you turn the water off, even if someone opened a hot water faucet there would be no water pressure to push hot water out of the water heater (WH), so the WH will not need to fire up.

Once you turn off you main water supply you will want to open all the faucets in the house to bleed out what you can of water in the pipes to minimize the amount of water that come out of the toilet supply pipe when you remove the valve. You can turn it to the pilot setting if you want to feel more at ease, its easy and the pilot will stay on so no need to re-light it.

The boiler is a closed system, except for a one way valve that is designed to add water to the system if it runs low. It will not "empty out and run dry" if you shut of the main water supply for a short time. You can turn off the power to it if you feel you need to.

I'm reading that I need to shut off the water, then drain the lines, before making the repair.

Yes and no to draining the lines. You really only need to drain the lines enough that you don't flood yourself when you take out the fitting or valve.

All this generally means is to shut off the water and then open a faucet at or below the level of the pipe you're working on. You can verify this by flushing the toilet and seeing if there's any water trying to refill the tank. Once it's dry, you can do your repair with minimal water running out of the valve. You don't need to drain the tank, but if you accidentally pull too hard on the line going to the tank, you could damage or move the seal, causing water to come out. That's pretty trivial if the tank were empty, but a minor catastrophe if it's still full.

With a boiler set up, and even on some tankless heaters, there's often a valve on the out-going side that you can close so water isn't drawn out while doing work.

That said, if this is lower than any other valve or faucet, you'll still need a pan to catch water that comes out when you remove the valve. Even if you do get all the water out, you'll still want to have a towel to catch the drips and drops remaining.

Also, because this is a toilet, it's taking water from the cold water side, not hot, so you shouldn't have to worry about the water heater at all.

The reasons for shutting off the water heater are twofold:

First, if the heater comes "on" while you have things disassembled then the water in the tank will expand and may cause some minor flooding.

Second, and more importantly, if something goes awry and you have to make a run to the hardware store, or some other issue drags things out, the tank may, for any of several reasons, begin to empty out. If the heater comes on with the tank partially empty the heater may be damaged.

It's a simple matter to turn the heater to pilot or turn off the power to it, so why not do it? Biggest danger is forgetting to turn it back on until the next morning's shower.

I did this myself recently and it was fairly straightforward. Granted, I have a tankless water heater in a single story home, but I wouldn't be worried about your water heater. The entire time to replace one cutoff valve is maybe 15 minutes tops, ten if you're using a push-on connector (mine were sweated on so I score-cut them off and replaced with Sharkbites). It looks like your connector is threaded, so it shouldn't be a problem. Remove your supply line and open the valve to let it bleed out whatever water it needs, and then remove the old cutoff.

Your water heater stores multiple gallons of water inside and that water is not near boiling. Most water heaters cap out at 140F (or 60C, which is a dangerous level to have it at). You're going to lose some water to attrition over time if you drain the lines, but we're not talking enough to make a huge difference. It would likely take days for the water to evaporate into the empty lines.

Even if it did happen more quickly, modern water heaters have pressure relief valves. It's typically a toggle valve on the side of your water heater. Put a bucket beneath the valve and carefully open it until water comes out (the water will be hot because that's what's inside). This verifies the valve is working.