He died (probably).
According the the script for Kung Fu Panda, Tai Lung got himself Ka-Thoomed.
Since it happened off-screen, I think we can be reasonably sure that he died, rather than experiencing any mystical conversion to a spiritual plane of existence.
TAI LUNG: You're bluffing. You're bluffing! Shifu didn't teach you that.
PO: Nope. I figured it out.
He flexes his pinky...
PO (CONT'D): Skadoosh!
[EXT. VALLEY OF PEACE]
A mushroom cloud appears over the Valley, sweeping past the Furious
Five and the fleeing villagers.
Notably, in the follow-up special "Kung Fu Panda Holiday", when Tai-Lung appears in a dream, Po states that he was sure that Tai Lung was dea[d].
And in the Legends of Awesomeness series, Po is pretty explicit about having killed Tai Lung.
Po: Hmm… Tai Lung, did you say? Hm, yes. Sounds familiar.
Peng: They say you destroyed him. But that… that can’t be true. You’re
my hero, Po. Please. Please tell me it’s not true.
Po: Yes, I did it! I skadooshed your uncle!
Master and the Panda