I've read some people writing that they've seen people be caught sitting in First/Business class when they have economy class tickets. This was for international flights on big planes with several hundred passengers.
Is this even possible, or is it improbable that you won't get caught? (At least when food is served)
And are there any consequences that it might entail?
There's no need for a discussion on ethics.
Usually flight attendants assigned on first and business classes make a quick head count after boarding is done. You can pass this if the class is almost full. In addition to that there are some techniques used to make sure that only people with the right boarding passes sit there, that's why there is always flight attendants in first and business classes during the boarding time. You might think they are there to help you sit or to help you with your jacket, that's true but they are also there to check your boarding pass either by asking you directly or by looking at it while it's in your hand (many airlines have different boarding pass colors for different classes). Also in many airlines flight attendants are asked to check the passengers names before serving to make them feel welcomed, if your name is not in the seat plan then you are busted.
Out of personal experience as a cabin crew, with some luck there is somehow a chance of succeeding in sneaking into first or business class even with all what's mentioned above, especially in big planes and when first/business class is almost full.
What are the consequences? well, the maximum would be paying the difference between economy and first or business class and big embarrassment. Again, out of personal experience passengers always say that they thought this is their seat (AKA playing dumb) and we can not do anything other than sending them back to their original seat or sometimes leave them continue.
Finally, if the same door is used to board passengers from all classes then your chances are higher and you playing dumb if they catch you will be more acceptable.
On all of the airlines I've ever flown I'd say it's extremely unlikely that you wouldn't get caught. The one exception might be on smaller flights where business class is little more than economy with slight better service (eg, many intra-Europe flights).
Ignoring the possibility of a fellow passenger reporting you, or a flight attendant actually seeing you move, the main reason you're likely to get caught is simply due to the level of service provided in Business/First class. Flight Attendants normally have a list of all passengers sitting in Business/First class, and will use this list for everything from simply addressing the passenger by name through to using it to collect meal requests, keep track of what passengers are drinking, etc.
If a seat is occupied that is not is not shown on their list then you can expect they will very quickly start asking questions as to why, and ask to see your boarding pass or some other identification they can use to check against the passenger manifest to see where you are supposed to be seated. At that point the best you can hope for is to be move back to your original seat.
That's not to say that doing this can't work - and I'm sure at least some people have done it in the past - but the odds of it succeeding are extremely low, at least on a medium/long-half flight.
I had an experience somewhat similar to what you asked about a few years ago.
During the flight from New York to Zurich, Dec 2011, I was seated in the Economy class, left row window seat, where the leg space is close to nonexistent. Being 7' tall (212 cm) I was in for 8 and some hours of flying in a very unpleasant knees-to-my-chest position and didn't look forward to it.
Half an hour after the plane got off I waited for the personnel to exit the cabin going on their various duties and moved to the first row of seats in the center (the one usually charged extra due to the extra leg space it provides). After the steward walked past me two or three times within the next 10 minutes, he rightfully came to the conclusion that I was not supposed to be in that seat and approached me asking (quietly and very tactfully) if the seat I am sitting in is mine.
I explained to the steward as best as I could that I was having a really bad time in the seat that was originally assigned to me, and that due to the entire row (along with the whole center block of seats) being mostly empty I sincerely hope I would not be causing any discomfort or unpleasantness to the few passengers sitting relatively close to me.
I guess being sincere and polite helped, for after explaining me that he was required to charge an additional cost for the premium seats because a) that was his job, and b) if he didn't then everyone would relocate to the better seats, he asked me to give him a "credit card" (which happened to be my library membership card) from which he theatrically "deducted the additional cost" of my new seat, handed it back to me, wished me a pleasant flight and continued on his regular duties. I finished the flight in my new and much more comfortable seat.
TL, DR: As long as you are well mannered and have a sensible explanation for your request, there is a good chance that your needs will be met without much hassle, as long as you pay good attention that, while doing so, you don't put the person helping you in an uncomfortable situation. They are just doing their job, and it's in their best interest that you have a nice flight and have the urge to fly with them again. If they can help you, while not exposing themselves to scrutiny, in my experience they often will.
There are legal consequences. This is considered "Theft of Service" and is a criminal act. It is no different from running out on a taxi or a haircut, or jumping a turnstile in the subway without paying. Odds are you will be caught as well. Most of the time they will just send you back. Reality is most flights these days are full, and FA's know who is supposed to be sitting where.
