What role do the telescreens play in the novel? How do they influence Winston’s actions?

The telescreen has the general effect of driving everyone, including Winston, into a state of emotional and mental disarray. At least through the daytime hours, the telescreen never stops blaring, and there is no way of turning it off. (Later, Winston discovers that the Inner Party members such as O'Brien are the only ones allowed the privilege of shutting off the telescreen.) Since the telescreen both transmits and receives, it is impossible for Winston and all Party members to do anything that is not being watched, so Winston is in a constant state of fear that he might inadvertently give himself, and his heretical thoughts, away. The anomaly of his flat is that there is a culvert or niche where he can stay out of view of the telescreen, and it is here that he writes his diary.
Though the Party intends the telescreen as a device of total control, including mind control, it actually does the opposite with Winston. The constant stream of nonsense that comes out of the telescreen, such as the falsified statistics about the production of goods and the "glorious" victories Oceania is achieving in the "war" against either Eurasia or Eastasia, exacerbate Winston's rebelliousness. It's partly because, in his job in the Ministry of Truth, he is engaging in the falsification of documents himself that he realizes the propaganda coming from the telescreen is lies, claptrap. And the obligatory exercise program in the morning increases his resentment. But the most significant fact of all is his knowledge that because of this ubiquitous device, he cannot do anything without the voyeuristic mechanism of the Party seeing and hearing it.
The fact is almost casually stated at one point that "most of the proles did not even have telescreens in their homes." This accords with the Party's ideology that "the proles are not human beings." Oddly, though, it also gives the proles the freedom from being spied on that Winston and others of his status lack. His affair with Julia has to be conducted in places where they can assume there are no telescreens. This is why they must first go to a remote rural spot and then take a room in a slum quarter, not knowing, of course, that the man who rents the room to them, Mr. Charrington, is actually an agent of the Thought Police.
The rural world is presented as the opposite of the poisonous urban environment, as Orwell viewed the real London, even in the peaceful time of the 1930s. Yet this Eden-like countryside turns out to be an illusion. And even without the telescreen watching, Winston knows that at some point his disobedience to the Party will be found out, and he will be punished.
When we see Winston at the close of the novel, after he has been "re-educated" into a Big Brother-worshiping zombie, the telescreen and the news it pours forth are now the only thing he has to live for. He drinks glass after glass of the cheap gin in the Chestnut Tree Cafe and listens and watches eagerly for news of Oceania's victories in the field, paradoxically realizing, as he does so, that the "longed-for bullet" is now entering his head. In being reeducated, he has become one of the dead, just as he and Julia said to each other in their final moment before being arrested by the Thought Police.