My story takes place in a city during a hard winter: streets are covered with snow, the wind is blowing, the river is frozen. I describe all the setting in the very first scene. My character leaves his coat behind in a closed pub and later suffers from cold in the streets on his way home.
How many pages could a reader remember it is winter? Should I remind him of the setting each time my character leaves his appartment?
Each time it makes sense, no more, no less.
You do not need to make recalls. Consider your character's POV. When does he think about the weather ?
Closing the door of my house, I looked at the sky. It will not be long before it starts to snow again.
or
Walking on the sidewalk, he was careful not to slip.
or
He looked at the landscape, beautifully wrapped in a coat of snow.
If weather is meaningful for your story, that should naturally show in concrete facts, not arbitrary reminders.
+1 Stephane. My own take is that if you are mentioning something like the weather, an emotional state, an article of clothing, a weapon, anything, it should have consequences in the story.
So yes, describing winter in your story does have consequences, a forgotten coat creates some hardship, a form of conflict that sustains interest in the story during a transition from pub to his home, that you chose to describe for some reason: perhaps a little world-building, or time to think or see something that has consequences later in the story.
However, all the consequences of winter can follow without further describing itself. There is a maxim in computer coding called DRY, meaning Don't Repeat Yourself, which applies to writing as well. Keep your story DRY!
If you are going to repeat yourself and keep telling people it is Winter, then each time you do that should have some consequence. Otherwise, you are treating the reader like a child by constantly reminding them of it. Tie those description to something that happens; even if the something is pretty minor, but if nothing else it can cause sensory or emotional reactions.
So YES and NO: I wouldn't mention it is winter every time he leaves the apartment. But the fact that it IS winter surely changes his behavior. He has to bundle up for winter; he is careful descending stone steps, he wears sneakers and thick socks, and keeps his office dress shoes and socks in his briefcase or a satchel. He has to walk around a dirty snowman on the sidewalk, he is wary of the icicles hanging off the porch overhang, and uses his umbrella to knock a few loose. Winter, ice and cold temperatures, should affect behavior (for everyone) and emotions; be it irritation or joy.
Pick some concrete consequence of the weather and describe it. "Winter" alone is vague and not concrete, you make it concrete through the specific conflicts or opportunities it inflicts or presents, respectively. Don't leave it up to the reader to guess what Winter is like for your characters, show them what it is like by how they deal with it.
Ditto to Stephane and Amadeus, let me just add:
I think there's a difference between blatant reminders and subtle reminders.
Using your coat example, I see three ways you might say this.
(a) Blatant reminder. "Jack walked out of his apartment wearing a light shirt and sandals. It was still winter. This meant that Jack should have been wearing a coat and so he was very cold." Well okay, I've exaggerated the goofiness of it, but you get the idea.
(b) Other extreme, not mentioning it at all. "Jack walked out of his apartment wearing a light shirt and sandals." Period. If the reader has forgotten that you said it was winter, he may not realize the problem. Or if the reader remembers, his reaction might be, "But wait, I thought it was winter. Is it spring now? How much time has passed?"
(c) Middle ground: bring up what's relevant. "Jack walked out of his apartment wearing a light shirt and sandals. Once he stepped out the door he realized this was foolish: the bitter cold sliced through his thin shirt."
In general, in real life there would be all sorts of aspects of the environment that are not relevant to the story, and which therefore should not be mentioned. I wouldn't describe what color the shoelaces of every character are, or the names of all the shops that a character passes as he runs down the street trying to escape from the kidnappers. (Of course some seemingly trivial things might be relevant to set the mood, or as red herrings in a mystery story, etc.)
There are three perspectives to this question, and environment or setting in general.
First, the reader/writer perspective: You can mention it whenever you deem it is necessary to remind the reader. The more different from the readers ordinary world the setting, the more often this will be as the mind tends to drift towards normalcy. Typically, you casually toss it into normal descriptive sentences, e.g. instead of "he walked down the road" you would write "he walked down the icy road".
Second, the story/meaning perspective: Mention it whenever it is relevant to the story or affects what is going on. Typically, this means the more different from the worlds definition of "ordinary" the current circumstances are, the more often it will affect something and needs to be mentioned. You focus on the setting or affect, e.g. instead of "he was chasing him down the alley, around tight corners and through the park", you would write "the snow and ice made the chase difficult, which went down the alley first, then he almost slipped at the tight corner and finally ended up in the park, trees covered in snow around him. In ordinary weather, he would have caught the guy, but in this coldness, he managed to slip away."
Thirst, the character/emotional perspective: Mention it whenever it would be on the mind of your acting character(s). The more outside the characters ordinary experience, and the more immediate, the more often. Extreme cold, for example, is not something you easily forget about when you are outside. The character would be constantly cold, uncomfortable, feet and ears freezing. As it is in the characters mind, you would write it in the story.
Remind the readers through action.
This is what vivid writing is all about, in my opinion.
If you do it well, there is no need for "descriptions" as such. You don't stop to talk about what the scenery is like. You weave it into your telling of your story.
The pub door banged shut behind Jack and he stood seething in the wintry air, laughter still ringing in his ears. Thrusting his fists deep into his trouser pockets, he slouched homeward, slipping occasionally on the slick ice patches that lined the sidewalk. Belatedly he recalled hanging his coat by the warm pub fire, but the thought of going back for it now made him cringe. He stiffened his back against the chill wind and marched resolutely onward. He'd show them, all right. Already ill-conceived plans and dire threats were bubbling through his mind.
Things are happening. Things don't stop happening and just sit there while you describe them. Instead, you should describe the things that happen in sufficient detail to coax vivid imagery to mind.
(Of course there are other schools of thought, such as James Joyce. :) But that sort of writing hardly could be said to have a "setting" as such, so your question wouldn't really apply in the first place.)
The real question here is:
If, as in your story, your character leaves his coat behind and wanders the streets in winter, he will probably feel the cold, and get colder with every minute. So the fact that he is "in" winter, will matter very much to him and he will spend much of his attention on it. Consequently, your writing should reflect your character's preoccupation with the wintry cold, slush, etc. The reminders to your readers will come naturally with the focus of your character.
Sure, it's winter and sure, it's an important part of your question, but is winter an essential feature of your story? Does the presence of cold have thematic consequences?
The main principle here is: refrain from describing things that have no importance to the story and its themes.
When you remember this, all else falls into place as naturally as warm soup into an empty stomach.
