Kloekhorst's Etymological Dictionary of the Hittite Inherited Lexicon proposes that
/u/ are separate phonemes in Hittite, but that the distinction doesn't appear in all environments: it's almost exclusively restricted to the position
As evidence, he analyzes plene spellings of
/u/ in interconsonantal position; the vast majority of words are spelled consistently with U (ap-u-un "that", mu-u-ga- "invoke", lu-u-ri- "disgrace") or consistently with Ú (lu-ú-li- "pond", ku-ú-ša- "daughter-in-law", ta-pu-ú-s°- "side").
Other evidence for a distinction comes from the verbal prefix u- (<*h₂ou-) "hither". On some verbs it's spelled with U, on other verbs with Ú, but the choice of one or the other seems to be consistent for each verb: u-un-na- "drive this way" consistently has U, while ú-da- "bring this way" consistently has Ú. The distinction doesn't seem to be phonemic in this environment, but it's nonetheless shown consistently in spelling.
His conclusion, from 18.104.22.168:
From the treatment above it is clear that the signs U and Ú, which are both traditionally interpreted as -u- only, in fact can be used to represent three different phonemes, namely /u/, /ū/ and /o/. Note that I do not distinguish a fourth phoneme, /ō/, for several reasons. First, the fact that the spelling of /o/ automatically requires the use of a plene vowel, namely the sign U, makes it graphically impossible to distinguish between a short /o/ and a long /ō/. Secondly, it is likely that /o/ behaves symmetrically to /e/, which does not show a phonemic distinction in length: when accented, /e/ is phonetically long in open syllables and monosyllabic words, but this lengthening is automatic and therefore subphonemic. I assume a similar behaviour of /o/.