A few posted by George Aaron Broadwell to sci.lang back in 1991:

A Zapotec tongue twister (dialect of Mitla):

Tsu'un buin zu^n kayu'n dzu`n kato'p bidzui`n.

'Thirteen men from Tlacochahuaya work with honey.'

Two Choctaw tongue twisters:

Chinchilla chikchiki chi_chokcolilaachi_h.

'I'm going to tickle your speckled chinchilla.'

Sho_shi showa showaka_ ishashshowaho_?

'Do you smell the stink bug stinking?'

[Phonetics: underline after a vowel means it's nasal. ^ is falling tone,
`is low tone, ' is high tone, is a high central vowel.]