In the abstract and en masse, class distinctions in Britain are acknowledged and of wide concern. Media and politicians are always talking about "middle class people", "working class people", "a growing underclass", and so on. These are general terms. There aren't specific recognised definitions of classes within which every person can be formally placed, although various social studies do use particular definitions. Politicians themselves find that their habits and backgrounds are constantly scrutinised and assessed for anything that conceivably might mark their social class. Other kinds of celebrity not so much (although it happens, for example Benedict Cumberbatch got into a brief row with some paper about whether he's posh). Normal people even less.
The words "toff" and "pleb", referring directly to social class and the behaviours associated with class, are both generally understood as insults. Recent coinages such as "chav" are invented to express specific aspects of the contempt people feel for particular social classes. In most contexts it's undesirable to be seen as "upper class" almost to the point of being an insult. There's some dismissal of "middle-class concerns" even by the middle class itself, from house prices to the quality of Waitrose humous, as being slightly embarrassing exercises of privilege.
People socialise to a large extent within their own class, although this is an observable tendency rather than an explicit intention or any serious social rule. I don't know whether Canada would be any different if you substitute "income and education" for "formally recognised socio-economic group", but whether it's unique to Britain or not it does add up to level of intolerance and disrespect between groups that occasionally is ugly.
So there's potentially an extremely long list of "do's" and "don'ts" here which add up to, "don't exhibit a stereotypical characteristic of one social class while surrounded by another social class". It's not a faux pas as such, but it would make you stand out. There are vast numbers of such stereotypes and fashions change, so it's hard to make a list. You'll know you've done it when you tell a crew of builders in a pub that your favourite drink is a particular Tuscan red wine and they all laugh (excessive interest in either wine or Tuscany being stereotypically middle-class), or you mention to a group you meet at Glyndebourne that you're thinking of going to Torremolinos (a stereotypically mass-market destination) and they look slightly pained. There's no guarantee that any particular group will sneer at preferences they consider unusual, British people are far from universally unpleasant about it. And as a tourist you'll never see these people again anyway, so you don't need to care. But it's about as far as the class system manifests that a visitor will encounter.
On an individual basis, the distinction you might encounter is between "common people" and "posh people", and even that is subjective. Brits won't specifically peg each other as members of a particular social class except as part of a general assessment of whether they're like each other or not. So, if someone you meet who considers themselves a working person establishes that you live on the income of a vast inheritance, they might choose not to chat to you in an amiable way about the trouble people have with landlords. You can call this a class divide, and it could affect the way people react to you as a tourist trying to strike up a conversation, but of course such judgements and differences would still exist in the absence of recognised social classes.
People fit into this system on the basis of their broad circumstances of wealth, occupation, education, official titles, and to some extent birth. Leaving aside the titled nobility, probably the strongest of these is education, in particular private vs. state education and going to university vs. not. That is however just my opinion, I'm no expert in the subject. Social studies that talk about class usually use occupation as the primary marker, with education a separate social concern. But popularly, private education can be a touchy topic in the UK and is strongly bound up with class. That doesn't necessarily mean that privately-educated foreigners will be viewed the same as privately-educated Brits. If you went to anything that looks like a British boarding school, then be prepared that some state-educated Brits would give you some stick for that if you mention it to them.
Anyway a foreigner who is the son of a foreign king would be considered of the aristocratic class, albeit not our aristocracy. A foreigner who is a medical doctor would be considered of the upper professional class, unless their circumstances dictate otherwise (I don't know, hypothetically an immigrant with medical qualifications working as a taxi driver). A foreigner who mentions they went to a recognised top university (or any university at all) might be rated a bit posh by a group of people with no thought of university themselves. A foreigner acting as a tourist isn't really showing many signs of their occupation, education and so on, and might therefore evade judgement.
No establishment will let you in or turn you away directly because of social class. What's important is you meet the dress code (if any), can afford to pay, and don't do anything grossly offensive. In the unlikely event that you do get thrown out of somewhere for gross offence, you might be thought of as low class after the fact, but I don't think that's fundamentally different from thinking of someone as "being ghetto" / "white trash" / whatever in the absence of a recognised class system. One thing that might be class-related is when the place is over-booked. Showing signs of high social status or connections might (not necessarily) make it easier to get a reservation in some cases. Being clumsy about it probably won't, so it's not worth getting on your high horse.
If a British person asks you what you do for a living, this isn't as such because they're trying to peg your social class. It's much the same as anywhere else in the world, they're either trying to get an idea what sort of life you live or they're looking for something to make conversation about. My grandfather, when he was assigned to a unit in the second world war, was asked by his commanding officer what his father did for a living. This was an important matter for class distinctions at that time, and my grandfather believes he was assigned duties on the basis of the answer (which as it happens was skilled working class: his father was a blacksmith, and this could well have ruled my grandfather out of immediate consideration as an officer). It's not like my grandfather is deeply bitter about it, he was only in for a few years, but you can imagine the effect on those whose desired careers judged them that way, not so long ago that they aren't still around in Britain in large numbers. That type of class discrimination really doesn't go on any more in any official way. Individuals might have their private prejudices.
There are also situations in which British formal protocol and etiquette is informed by the class system, especially if you're planning to meet royalty or the aristocracy in a formal setting. If that happens, which it certainly won't if you don't want it to, there will probably be someone on hand to advise you. Ignorance is expected and catered for, but wilful disregard for what you've been told is rude. Most obviously you'll want to know what to call everyone: when you first meet them, in conversation and when referring to them in the third person. This is all about titles and only applies in formal settings or letter-writing. If you should chance to meet a vicar and a Duke in a nightclub, then both would pretty well certainly want you to call them by their first names in conversation, not "Reverend" and "Your Grace" respectively.
It's true that knowing what neighbourhoods to avoid can be tricky, but generally you'll do this on the basis of street crime and the issue of whether a stranger has any business wandering around a particular residential area, the same as any other country. If you don't go into a marked housing estate that is presumed mostly lower-class occupied, then it's for the same reasons you wouldn't go into a housing project in the US, not because of formal social class. If you don't go into a gated community (and there aren't many in the UK) then it's because you don't have an invitation from a resident, not because you aren't upper class. If you don't go into a high-end restaurant, then it's because you don't fancy the formality or the expense, not because they'll check your family tree at the door.
Be aware that especially in large cities and public spaces, British people can be quite reserved and dismissive. This might feel like they think you're the wrong social class, or you've unknowingly made some obscure error, or that Britain is very class- or protocol-bound. Typically they just didn't fancy talking to a stranger, and there's no social expectation they should have to if they don't want.
In short, you really don't need to know anything much about the class system to visit Britain. You might need to know a bit to follow some of the politics and the comedy.