No. Diablo III is a standalone story. You can jump in without playing the previous games.
Why It's Standalone:
New Protagonists: You play as a Nephalem hero (Barbarian, Wizard, etc.), not the Dark Wanderer or Heroes of D1/D2.
Self-Contained Plot: The story (defeat the Prime Evils, stop Diablo) is explained in-game via lore books, dialogue, and cinematics.
Modern Systems: Skills, runes, paragon levels, seasons, and Adventure Mode are unique to D3.
But Playing D1/D2 Enhances the Experience:
Lore Depth: You'll recognize names (Deckard Cain, Tyrael, Adria, Leoric, Lazarus), locations (Tristram, Kurast, Pandemonium Fortress), and the history of the Soulstones.
Atmosphere: D1's dread, D2's gothic grit, and D3's cinematic action show the series' evolution.
D2: Resurrected (2021): Faithful remaster on PC/console—best way to experience the classic.
Recommendation: Play D3 first if you want modern ARPG gameplay. Play D1/D2 (or D2R) later if you fall in love with the world and want the full history.