I can't think of any and google has not been helpful.
Voiceless vowels are quite possible, and occur in one way or another in many languages.
After all, all vowels and all consonants that are whispered are ipso facto voiceless.
Whisper [a] and you have pronounced a voiceless vowel.
However, the overwhelming majority of vowel sounds in speech are voiced, since vowel formants are modifications of a voiced airstream from the larynx. Exceptions to this rule fall into a number of categories.
- in some languages, like Japanese, some vowels become voiceless in some environments
(in Japanese, high vowels /i/ and /u/ are devoiced between voiceless consonants)
- in some languages, like English, voiceless vowels are allophones of a consonant phoneme
(English /h/ is a voiceless vocal onset, a voiceless version of whatever vowel it precedes)
- in some languages (Acehnese, for instance) some vowel phonemes are contrastively voiceless
(this is quite rare, however -- most voiceless vowels are conditioned rather than contrastive.)
Yes, it's technically possible, but I think you'd only consider it as an allophone, otherwise, the better transcription might be /h/. Wikipedia lists one example in English as potato, where the first o is not vocalized, but otherwise the mouth is in the same position and there is air flowing. (IPA transcriptions use a ring for devoicing).
Japanese seems to the more common language for it with their u vowel, but thinking about, European Portuguese might well do it where traditionally transcriptions will normally list them as elided.
For an academic citation, here's an article on devoiced vowels in English and German: "Vowel devoicing/deletion in English and German", by Jonathan E. J. Rodgers.
To add to John Lawler's answer (since I can't comment): Shuar (a Jivaroan, or Chicham, language of Ecuador) has (or did have a few decades ago) voiceless vowels in certain positions. I'll quote from Glen Turner's 1957 IU dissertation: "In a contour [= phrase] final open syllable, a single, unstressed, oral vowel following a consonant cluster of nasal plus homorganic stop or affricate or following any single consonant other than /w/ or /y/ may be unvoiced if it is also preceded by 2 or more vowels in the same word... When followed by a morpheme or another word (i.e. contour medial), voiceless vowels may be come voiced, lost, or remain present but unvoiced..." This word-final position was post-consonantal, so it would be odd to consider it to be an /h/.
FWIW, in English I usually pronounce 'Mississippi' with a voiceless vowel in the second syllable. Again, given the position, it would be odd to consider this to be an /h/. Otoh, it is more or less indistinguishable from lengthening the 's' (that is, merging the first 'ss' with the second).