The short answer is no. The Nintendo DSi cannot play Nintendo 3DS games—period. This isn't a firmware limitation or a missing update; it's a fundamental hardware and architectural incompatibility. Here's a detailed breakdown of why the DSi falls short.
To play 3DS titles, you need any system in the 3DS family: the original 3DS, 3DS XL, 2DS, New 3DS, New 3DS XL, or New 2DS XL. All share the same core architecture and cartridge slot. The "New" models add a faster CPU and extra buttons (C-stick, ZL/ZR) required by a handful of exclusive titles like Xenoblade Chronicles 3D, but they play the full standard 3DS library.
The DSi remains a great system for its own library—DS, DSiWare, and Game Boy Advance (via Slot-2 on earlier models)—but it's a dead end for 3DS software.