As I recall, there is a very thin, plate-like, plastic nut (maybe 3-in across) located at the center of the underside of the toilet tank to hold the drain tube in place. Years ago, I had to replace those components, and it was a bear because no typical tool would grip the thing. After the fact, I seem to recall hearing of something called a flange wrench, is that correct? Is that what I need to grip it? Please advise.
I usually use a big pair of channel lock pliers . My 16" pair handles toilet fittings. It’s pretty useful for lots of other things, too.
<sigh> I read that as “flange wench”… took me back to my university days </sigh>
You’re not alone. I read the thread title as “What kind of wench do I need for …”.
My expectations, not for the first time, have been sadly punctured. 
I usually use a big-ass pair of channel locks.
That’d be to small for anything that you would use a really big channel locks for, it’s used for loosening the nuts on pvc drain pipes.
A Strap wrench might work.
No, this is a really large wrench and is used for exactly what the op is asking about. It’s a large, thin flare nut style wrench. It would be used when it’s tough to get a large pipe wrench on it due to space and size of the nut.
I have a 2 piece toilet with a chrome tube connecting them and that is the wrench I used.
FYI, if you have a 2 piece toilet and are replacing the chrome tube I found an easy way to keep from ruining the tube when cutting it to size. I put a piece of PVC pipe inside it of the same diameter. This worked even after cutting it and bending it out of round. I found this out after mangling the pipe after cutting it. The insert is still there today to hold it’s shape.
On all of the toilets I’ve replaced guts in, I don’t think I’ve ever used a tool on that nut. I just get it a good hand-tight, and none has leaked.
How I do it is to get it snug as possible while turning the nut with my hand, then the final snug is done while holding the nut still, and giving the overflow / flush valve a final twist. Of course, it is possible that my hands are simply bigger and stronger than yours.
Both of my potties (I have a 2 year old so I forget their real name) have 4 large tabs on those nuts so you can tighten them by hand.
My fault, I read the OP as trying to remove a nut somewhere between the base of the toilet and the closet flange (I don’t remember one being down there, but I’ve only replaced one toilet), so I was thinking something that would be as big as the waste pipe. Then someone mentioned a 16" channel locks, so I put those to together and that made the plumbers wrench way to small.
Nevermind.
(Nitpick, if the OP guessed the size of the nut correctly (3") the linked plumbers wrench would be to small as it only goes to 2.5")
While you are correct it should be hand tight, it is (a) too flat to grasp (even with convention wrenches others have suggested, and (b) it has tightened in place with age. Again, I am 99% certain there is a special, crescent-type or box-type wrench to “hug” this item which is simply too flat to grab any other way.
Before I go looking foolish at the hardware store, let me look foolish here! 
Doesn’t a flange wrench sound familiar to anyone? Any other thoughts on this?
I have never heard of a flange wrench. And I buy a lot of tools. A medium pair of Channel locks (8 inch or so) is all that is required for this job.
I just did this same job in my master bath. Here is what I would suggest.
Look at the angle stop (shut off valve coming out of the wall or floor) is it threaded or is it a compression fit onto copper or plastic pipe.
Buy a new angle stop of the same style, get the 1/4 turn type, they don’t wear out like the type with faucet washer do.
Buy a new toilet supply pipe, get the steel braided flexible kind, most likely 9" is plenty long enough.
Buy a new ballcock (yes that is what it is called. Call it that at the hardware store and you will be labeled as a pro 
)
Double check that the supply pipe fits the outlet of the angle stop before you leave the store.
Turn off the water supply to the house. Flush and hold the lever down until the tank drains.
Armed with your channel locks, remove the toilet supply pipe. Remove the angle stop, and replace. If it is threaded, use pipe dope, or Teflon tape on the threads.
Place a small pan under where the ballcock enters the tank. Again using your trusty channel locks loosen the big plastic nut holding the ballcock in place. NOTE: If you are sitting backwards on the toilet and reaching down to spin the nut, you will turn it clockwise from this point of view.
Once the nut is loose enough to turn by hand, put down the channel locks, push down on the top of the ball cock and spin the nut off. Then pick up the pan and hold it underneath the ballcock. Lift up the ballcock and the remaining water will drain into the pan.
Install the new ballcock, make sure it is oriented so that it does not interfere with the flush valve, or tank lever. Install the overflow tube.
Tighten the nut good and snug with you hand
install the new supply pipe, the top end can be hand tightened, but you will need the channel locks for the bottom nut.
Turn the water back on and check for leaks.
The plumber’s wrench link was probably what you are after. It is also called a Spud Wrench. Here is a link which gives the sizes of nuts it will fit ( 1-3/4, 2 and 2-1/2") :
http://www.acehardwareoutlet.com/(evtjj0mmkbwsdvrjgc3y05r5)/ProductDetails.aspx?SKU=2130006274
This is the nut you are talking about, right?
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