Here’s a good post on Quora about the different types of sarcasm and irony used in Japan: https://www.quora.com/Is-the-usage-of-sarcasm-common-in-the-Japanese-language/answer/Isaac-Hsu-1
The author of that talks about 皮肉 as a more generic term for sarcasm, satire, etc. and 嫌味 (iyami) as a more biting kind of sarcasm.
He also points out (as others here have) that Japanese folks definitely have a sense of sarcasm, which matches with my own experience of living there briefly back in 2005.
As a haikuist, I’d like to point out that there are a lot of wonderfully sarcastic waka, senryu, hokku, etc. from at least as far back as the 1350s. In fact, the root word for haiku is 俳諧連歌 (haikai renga), which means roughly “popular or unorthodox linked verse”, and as English Wikipedia notes early haikai “often derived their humor from satire or puns.” Shirane Haruo has a brief description of them here.
How sarcastic are they?
My favourite example is from the inu tsukubashu, and is described pretty well in this web article.
the robe of haze is wet at its hem
Princess Sao of spring pissed as she started
This sounds just vulgar, but it’s subverting typical courtly poetry at the time, which described the hazy dampness of spring as being the hem of the Saohime’s robe, wet with her tears. (The wet hem of a robe being a stereotypical way of describing sorrow in waka.)
So I’d say that sarcasm/irony definitely is a concept with a long and varied history in Japan.
Like others here, I suspect the misunderstandings around sarcasm have more to do with cultural differences. We currently have a student assistant where I work who is Japanese, and because our work environment is, uh, a little sarcastic, she is often taken aback when her supervisor seems to be asking her to do something ridiculous. She understands that it’s sarcasm, but it’s so out of place that it takes her a while to parse it, every time.
Japanese Wikipedia, interestingly, redirects to either 嫌味 or イロニー (irony) when I search for 皮肉, and the page for 嫌味 seems to be a more catch-all phrase while the page for イロニー refers to the much more narrowly-defined meaning of the term as used by (classical Greek) rhetoric and (Kierkegaardian) philosophy.