Translation:We walk from the square to the cinema.
This is more of a suggestion than a question or pure comment: If you wanted to better support American English, this sentence would often be translated as, "We walk from the square to the movies." (A theater that shows films is often colloquially referred to as "the movies." See, for example, the song from the 1982 film "Annie" entitled "Let's Go to the Movies.") We typically do not often use the term "cinema" to mean the place where films are shown but rather as the study or art of film.
Technically, I am not sure if movies would have been rejected. I tried it before, and it was rejected so now I generally only translate "phictiúrlann" as "cinema." Again, just a suggestion; it's not necessary.
It's possible that Duolingo would have accepted "movie theater", but a "theatre" is where people go to see live actors perform a play, and amharclann is the Irish for "theatre".
On kjsoda's original point, the NEID does include "to go to the movies" - dul chuig an bpictiúrlann as an example of translating "to the movies" into Irish, but the closest idiomatic match in Ireland is actually "to the pictures", which gives us "to go to the pictures" - dul chuig na pictiúir.
Note also that, at least in some dialects, go dtí an phictiúrlann would imply going as far as the cinema, but not actually going inside to watch a film, whereas chuig implies going for the purpose of participating or attending.
Looking at this again, US English is losing it's synonyms as it continues in the process of becoming a trade language. In 1967 cinema, movie theatre et cetera would have been much more common than the present. Now you'll seldom hear anything except 'the movies' or just the name of the individual film being seen.
I'd guess in another 50 years British English will still be rich in synonyms but US English, will be more even more sparse than at present.
I appreciate the information on the other Irish phrases.
Your first paragraph leaves out the bit I was commenting on, though. For Americans, a theater isn't just the name of the place where you go see a play. It's also the place where you go see movies. The first entry in dictionary.com includes movies as well. It just seems weird to me that it's still not acceptable as an answer.
I did respond to the bit you were commenting on - I suggested that "movie theater" might be a useful alternative answer, but that "theater" on it's own could be confusing.
I just went to Bing.com's news search, set it to the US, and searched for "a new theater". All of the results on the first page referred to "performing arts" type theaters, except for one from a business magazine discussing a business that manages movie theaters.
Then I did a general search for "we went to the theater", and in fairness, there were some hits that clearly referred to going to a building called a theater to see a movie. But they were a small minority of those responses, and were clearly dwarfed by the number of hits for "we went to the movies" (dozens versus thousands).
You're right to suggest that the task of translating a text does require a degree of cultural adaptation, but that cuts both ways, and even in American English, "theater" is sufficiently ambiguous that it's a poor choice in this. Even some Americans would be confused by the use of "theater" in this sentence, whereas I doubt that "movie theater" would cause any such confusion.