A Community Manager post to the Battle.net forums clarifies Blizzard's nuanced approach to multiboxing in Diablo III. There's nothing to prevent players from having multiple free Battle.net accounts, they (unsurprisingly) do not object to paying for multiple copies of the game, and there are no rules against playing more than one character at once, but third-party software to allow the control of multiple characters at once is at issue, and using it could be enough to trigger a permanent ban:

For Diablo III, multiboxing walks the strange line of being something we technically allow, but don’t officially support.

For clarity, players are allowed to have multiple Battle.net accounts, and each of them can have a Diablo III: Reaper of Souls license (one per account). There is no limit to the number of Battle.net accounts a single user can have, so long as no more than three of them have Battle.net Balance*. There is also no restriction to how many of these accounts can be online at once, but the methods to do so may not be supported. We do not support third-party software, and third-party software that automates gameplay is not allowed. Accounts found to be using third-party software that automates gameplay risk being closed, possibly permanently.

In other words, you can have multiple accounts, and you can be online with all of them at one time. We don’t support multiboxing in Diablo III, however, and some software used to facilitate multiboxing may violate our policies, so do so at your own risk.

* Please note that this clarification is not reflected within the EULA and TOU at this time. We will be updating both documents in the near future to correct this.