In Fahrenheit 451 there is a connection to Puritanism in many ways. One big one is the gap between the government/authority and the people which is similar to the gap between God and the people. The people in the community in the book don'r have any idea about the government or what it does and its like this overarching power that has all these rules and set ways and it plays God in the society. In Puritan times they didn't know what God was they just believed in Him and trusted that He was responsible for everything and that all that He did they deserved. People in this book feel the same way about their authority.
Another connection is that Puritan society and ways were set to help people understand the society better. The simple jingles that play in the earpieces and around town and the soap operas that play on the televisions, the fast cars and no books or thinking for oneself all happen under the idea that a simple society in which if the people know less they are better off is all the government's way of making the citizens understand everything better. If something happens its because someone broke the law- they had a book or they asked too many questions, which the government and firemen don't like. Then there's the level of access-- everyone can get a television or ear buds and if not they hear the jingles in public places and there's a fix for everything, like when Guy's wife overdosed men came over and fixed her and she was none the wiser. This idea of people understanding what was going on as they chose to understand it and how the government wants them to understand it is similar to Puritanism and then the easy access to at least some influence of the ideas-- Puritans could just believe and thank God for what he did and that was enough and in Fahrenheit 451 all a citizen has to do is follow the crowd.
Just as Puritans feared and revered God, citizens in Montag's world fear and revere the authorities. The firemen and the men who come if you do something bad-- they don't want to get in trouble and if a firemen showed up at their door they'd freak out just as Puritans would freak out when someone would say that God wasn't real or something along those lines because they would get scared of what He would do-- same goes for how the citizens would be if someone they knew announced they had books, as Guy did to his wife and her friends. This book also connects to the idea of a city on a hill in that everything is seemingly perfect and it's all to make everyone's life easier and it all coheres with the beliefs most have about books and the societal rules. The idea that if one gets bored they can go speed race or be a part of a "family" or just call men over to fix the problem completely strives to put the image of a perfect society in the reader's mind if there were no downsides to it, such as are hardly seen by Puritans unless they look hard, in which case they are punished, like in Montag's society. One can get the problem fixes no problem no questions asked.
The last way in which Fahrenheit 451 and Puritanism connect is the idea of limited atonement. The only difference is that the ratios of those who were in good standing are switched- more are better off in Montag's society than in Puritanism. If one doesn't think for themselves and don't read or ask too many questions they are better off and wont get a second glance whereas one wrong question or a book in your hands will warrant trouble and punishment. In Puritan times the atoned were only around ten percent of the population and they were in good standing mostly because of predestination but they were still in good standing, as the citizens in Montag's society were for the most part.