As you can read on the bungie FAQ/Help: Yes, you can play it without PSN+ (corrected my first answer, sorry for that), but not access all features:
Q. Does Destiny require PlayStation®Plus or Xbox Live Gold?
No, PlayStation®Plus and Xbox Live are not required to play Destiny. However, PlayStation®Plus or Xbox Live Gold is required to access some content. Please see below for more information.
What is PlayStation®Plus?
PlayStation®Plus works differently for the PlayStation 3 and PlayStation 4 consoles.
PS4: PlayStation®Plus is required to join Fireteams and play some online activities, such as Strikes, the Raid, and the Crucible.
PS3: PlayStation®Plus is not required.
Xbox Live Gold
What is Xbox Live?
Xbox Live Gold is required to join Fireteams and play some online activities on the Xbox One or Xbox 360, such as Strikes, the Raid, and the Crucible.
Please note: Xbox Live Gold is required to download and play on Xbox platforms. 48-hour, 7-day, and 14-day Xbox Live Gold trials are not supported on Xbox 360.