Since The Heart Is a Lonely Hunter is set in the deep South, you're probably thinking there can't be much going on with this symbol aside from the fact that it's hot. And humid. All the time.
But there is something to that excessive heat, as well as the other odd weather episodes that occur throughout the novel. The weather is a key part of the setting and the local color, and it is definitely helps to set the novel's mood. Along with this, the weather also acts as a key Southern gothic component, and often lends scenes a sense of the uncanny and the bizarre:
Then when it looked like the rain would keep on and on forever it suddenly stopped. They woke up one morning and the bright sun was shining. By afternoon the weather was almost warm as summer. [...] The kids looked hot and sticky and their winter clothes had a sour smell. (2.5.2)
There's a sense of not-quite-rightness all throughout this scene, with the bizarre heat wave in the winter. All the weirdness with the weather helps set the stage for the upcoming scene, where Bubber accidentally shoots Baby Wilson. Prior to this, Bubber had said that "I come to believe we all gonna drown" (2.5.1) due to the epic rain. The rain hit almost Biblical proportions, but it stopped before washing all the sin out of the world. Then the heat struck, and so did tragedy.
The weather also acts as a foreshadowing device during Jake's final scenes:
The vivid green of the trees along the sidewalk seemed to steal into the atmosphere so that there was a strange greenish glow over the street. All was so hushed and still that Jake paused for a moment to sniff the air and look around him. [...] But he was not quick enough. There was one metallic clash of thunder and the air chilled suddenly. Large silver drops of rain hissed on the pavement. (3.2.44)
This is some pretty heavy-handed symbolism, don't you think? Basically, the second we stop to take a breath (i.e. good weather), something terrible (i.e. bad weather) is bound to happen.