How is weather or the season related to the action in Chekhov's "The Lady with the Pet Dog," and if Gurov and Anna's affair heats up in Yalta's hot sun, why doesn't it cool down in Moscow's winter snow?
The answer to the second part of your question is actually that the Yalta heat does not act as a motivator to action, thus the Moscow cold does not act as a contrasting motivator. So Moscow does not cool the romance because Yalta warmth did not heat it up. The reason behind this assertion is that Russian realism did not use settings as motivational elements to drive characters to actions. This is an element French existentialist writer Sartre used is The Stranger and the American playwright Tennessee Williams used in Cat on a Hot Tin Roof, but this is not an element used by Russian realist Anton Chekhov. You can also see Chekhov's approach in "Love": the "soft April night" provides psychological development, not motivational.
Chekhov, like Dostoevsky, Turgenev, and Gorky, uses setting to reveal the psychological complexities of characterization, not as an external motivator. Russian realists do use setting to demonstrate the assault of nature upon humankind, though the assaults do not represent external motivators. This is because Russian realists focus on internal psychological motivators that drive a character within themselves. Think of Dostoevky's Raskolnikov in Crime and Punishment: he is driven by psychological factors while he is assaulted by the rain that pummels him after his foul deed thus complicating his decisions. Think of Gogol's clerk in "The Overcoat": he is driven by psychological motivators while the snows and rains pummel him as he pursues the overcoat thus complicating his quest. In this sense. weather and seasons become antagonists.
This answers the first part of your question: The weather and seasons are related to the action in that they antagonize psychological conflicts by adding external assault, which is a complicating element. Having said this, if you were to take the view that heat should motivate the affair and cold should be expected to dampen it, is there textual support for this view?
Heat/Yalta: It's hard to make this case, but one might say that the hot weather is Yalta heats to a sultry frenzy, then in the relief of the aftermath, passions agitated by the wind and heat are redirected to carnal expression.
It was sultry indoors, while in the street the wind whirled the dust round and round, and blew people's hats off. It was a thirsty day, ... The wind had completely dropped, but Gurov and Anna Sergeyevna still stood as though waiting....
Cold/Moscow: Gurov feels most like himself in the cold snowy air of Moscow. He feels exhilarated and easily falls into his normal routine starting with the children's breakfast in the dark before first light. He has every expectation that Anna will fade like the embers of his fire as all other women of his dalliances have done. Yet her memory surprisingly grows stronger and seems to be with him every moment of every day. It's hard to make this case, but you might say that while the heat of Yalta ignited his passion, the steady fire of home's wintry hearth burned her image deeper in his heart while reviving his youth, therefore, cold did not dampen Gurnov's passion. [You can do a similar analysis on Anna's setting to find textual support for her continued love.]
When the first snow has fallen, on the first day of sledge-driving it is pleasant to see the white earth, the white roofs, to draw soft, delicious breath, and the season brings back the days of one's youth.