It's a pretty big operation. Between the studio, the actors, supplying the city, and so on, what would be the cost per year? Also, did they ever explain how they pay for the thing? That'd be a lot of Burger King commercials...
TL;DR; The show would probably cost around US$3.5 billion per year to operate, and gets its money from advertisements and product placements.
The money, as with all television shows, comes from advertising. Not only would they be able to get commercial money, but there were wonderful opportunities for product placement as seen here (at about the 0:20 mark):
There were lots of these throughout the movie, which would easily be enough to fund whatever they wanted.
We can also see some merchandising take place, as we see various items with his likeness:
In reference to how much it would cost, that exact question was asked (and answered) on Reddit, and the total was worked out to be about $3.5B/year.
I am unable to find the size of the dome via Google. Let's put it at 2 miles in diameter, and less than a mile tall (pictures don't indicate that it is a full hemisphere).
This would be the largest structure of its kind by far.
The current tallest structure is the Burj Khalifa at about half a mile high. The construction of a skyscraper is a bit different (and a more well-understood engineering feat... we've built other smaller ones for a century at this point). Its price tag is between $1 billion and $2 billion. This is not atypical, many other large scale engineering projects are in the single digit billions, even into the low double digits (Boston's Big Dig, etc.).
Then there's China's Three Gorges dam. This might be a better comparison, given its scale and constraints. The price tag on that is north of $20 billion, on up towards $30b depending on who you ask.
Given the novelty of the dome and its unique challenges, it's difficult to believe this isn't double digit billions in cost. Call the generous estimate as $10 billion.
So, if Truman is expected to live until 75, how long do they think the show will be a hit? Will people tune in to watch him wasting away in a nursing home? Dunno. But let's say they expect him to be lucrative for 50 years...
Even with a 50 year loan on that $10b, even if they get 1% interest we're talking something like $255 million per year in payments.
The cast (except Truman) is somewhat small. Lots of extras and so forth. So costs for them are probably comparable to some other network show.
But we have things like maintenance on the giant dome/stage. The size of the crews to do that work. Supplying an entire town with homes, cars, necessities. If 2000 people live there, what's the size of the economy itself there? Let's call that 600 households, at $100,000 per household/year. That's $60 million right there. And we haven't even gotten into the production crew yet (the people who script the show, as much as it is, producers, and the IT personnel who maintain all the cameras and the software to use them). This is probably several hundred people, minimum (a regular tv show today can have 100 if you count part-time people).
And I don't even know how to calculate the costs of the hand-wavey weather technology.
The cost for this show approaches (but maybe not exceeds) $1 billion per year.
However, I don't think this figure is realistic... we already have shows like Game of Thrones and Marco Polo at $100million/season, and Truman seems as if it should be another order of magnitude above what I've concluded (it can't be just ten times as expensive).
Though, this might be a matter of the innovation of amortizing a show's cost over 50 years, rather than paying year-to-year.