Update: Blizzard strongly denies that it banned players using Linux in conjunction with WINE to play Diablo III. Here's what they told Ars Technica:
"Playing the game on Linux (although not officially supported) and/or using Wine will not result in being banned, but cheating will. We’ve extensively tested various scenarios related to this situation, including replicating system setups for those who have posted claiming they were banned unfairly, and have not found any situations where players were banned solely for using Linux or Wine."
Readers have pointed out that our story was a bit confusing because Blizzard had never said that it considered WINE to be "Unapproved Third Party Software." The inference that WINE users were being banned for using it came from WINE users themselves and our own conclusions that – if players weren’t cheating and the only third-party program they were using was WINE, then perhaps Blizzard used false-positives to ban well-behaved users. See WineHQ for examples. Still, we apologize for the confusion and we hope Blizzard's latest statement brings more clarity.
Original Story: According to numerous complaints in this Battle.net forum thread Blizzard has permanently banned some users for trying to play Diablo III under Linux. According to one response in the thread Blizzard told him that the reason his account was banned was because he used "Unapproved Third Party Software."
Here's Blizzard's description of "Unapproved Third Party Software."
"A third party program is any file or program that is used in addition to the game to gain an unfair advantage. These programs may increase movement speed or teleport heroes from one place to another beyond what is allowed by game design. It also includes any programs that obtain information from the game that is not normally available to the regular player or that transmit or modify any of the game files."
Later on in the thread Blizzard Community Manager Bashiok offered the following explanation on the bans, noting that using Linux won't get you banned but using WINE will:
"We’ve extensively tested for false positive situations, including replicating system setups for those who have posted claiming they were banned unfairly. We’ve not found any situations that could produce a false positive, have found that the circumstances for which they were banned were clear and accurate, and we are extremely confident in our findings. Playing the game on Linux, although not officially supported, will not get you banned – cheating will."
So basically Linux is not supported but that won’t get players banned. But if you use it with a Windows emulator (WINE), then you are using "Unapproved Third Party Software." This does not make a lot of sense because the only way to play the game via Linux is with WINE…