"In the heat of the night" doesn't simply mean hot weather at night, does it?
Yes it does.
It is the title of a famous film based on a novel published in 1965. I am not aware of any prior or more general usage of the phrase.
From this it also suggests alterations in human behaviour when subjected to long periods of unaccustomed heat; shortness of temper, heightened emotions.
The phrase "the heat of the night" appears at least as early as in Hermann Hagedorn, "A Boys' Life of Roosevelt," in Boys' Life (November 18, 1918), in a scene set in Kenya:
At last the stars began to pale; to the heat of the night succeeded the more merciless heat of the day. Higher and higher rose the sun. The sweat streamed down our faces, and the bodies of the black men glistened like oiled iron.
One of the characteristics of the U.S. South—at least at lower elevations during the summer—is that the air rarely feels cool even at night. I grew up in southern Texas (average daily high temperatures in August, 92°F to 94°F; average daily low temperatures, 74°F to 76°F) and never realized how unusual this phenomenon was until I moved to Washington, D.C., in August of one year, and noticed that the air felt slightly cool in the middle of the night—although people there were complaining about how hot it was. To me, "in the heat of the night" memorializes the distinctive strangeness (from a Northerner's perspective) of nights that never really cease to be warm.
The plot of the 1967 movie In the Heat of the Night, which seems to have been responsible for popularizing the expression, involves a collaboration between a black homicide detective from Philadelphia and the white police chief of Sparta, Mississippi, to solve a murder case in Sparta—and I had always imagined that the title referred to the Philadelphia detective's experience of the weather. Sparta is located in the Deep South about an equidistant from Memphis, Tennessee (to the northwest), and Jackson, Mississippi (to the southwest).
As it turns out, however, the nights, aren't much hotter on average in Sparta than they are in Philadelphia—at least not during August. Average daily temperature highs and lows in Sparta during the first half of August are 92°F and 70°F and during the second half are 91°F and 69°F; in Philadelphia, the average daily temperatures historically range from a high of 86°F and low of 70°F on August 1 to a high of 81°F and a low of 66°F on August 31.
Even so, for many people accustomed to cooler, drier climates, temperatures that stubbornly and consistently remain in the middle 70s for much of the night, with high humidity, are a remarkably unpleasant thing and fully justify the expression "in the heat of the night."