The start date for the Black Ops 3 beta is August 19, as confirmed by Treyarch on Twitter. There was no mention how long the beta is going to last, but we'll have more information on that soon.
@Treyarch: "We're excited to announce that the #BlackOps3 Multiplayer Beta will begin on 8/19 starting on PS4. More information coming soon!"
As previously announced, the beta starts off as a PS4 exclusive. Xbox One and PC will have their beta at a later date, but it's yet to be revealed.