Boundaries
I think the source of the confusion here is that you think of the blob of NO2 gas as "a thing". Then, you see this "thing" doing spontaneous work against gravity, and assume: "Hey, <PERSON> says it should stay at the bottom, but it doesn't!" So there are two problems involved with the naive explanation.
The first is that the blob of gas is only "a thing" because of its container. The beaker is really the only thing binding the gas (technically, there is a trivial amount of gravitational attraction, but I hope we agree that this is many orders of magnitude too small to be of interest here). If you took the gas out of the beaker, by tipping the beaker over, you would very quickly see this gas cease to be "a thing", as it diffuses into the surrounding air.
What happened? Did the gas cease to exist? Of course not. The molecules of the gas just all went their separate ways. The only way to explain the dissolution of the gas blob is to consider it as a collection of molecules, each obeying <PERSON>'s laws to the extent of its accuracy (I hope we agree that the molecules are probably all moving non-relativistically here). That's what the other answers are trying to tell you. 185
The gas is not "a thing". It's a bunch of things. A bunch of billiard balls bouncing around very Newtonianly.
Balloons
The other problem is that work is being done, and a cost is being paid, and in any other context, you would not think anything unusual about the "transaction" occurring here. The quote clearly states that heat provides the work necessary to move the NO2 upwards against gravity. I'm guessing you have seen this principle at work elsewhere. Now, this isn't exactly the same case as the hot air balloon, because the NO2 is presumed to be at the same temperature as the air above at the start of the demonstration. But if you consider that the temperature of the mixed gases is lower than the pure NO2, then from that perspective, the NO2 is indeed "hot" and is rising due to this "excess heat", to put it simplistically. 185