Potential counterparts do not alternate because both of them are/​one of them is prohibited (or at least strongly discouraged) in non-initial syllables/​other (additionally long / reduced vowels)

Seto

/i/ : /ɨ/ is a potential alternating pair but /ɨ/ is prohibited in non-intitial syllables; all long vowels are prohibited in non-initial syllables

Erzya

First of all, it can be questioned whether /u/ : / i/ is a potential alternation (see Section 2.2.2.). If we consider that it is, the restriction of /u/ and that of /i/ are different. The vowel /u/ is practically prohibited in non-initial syllables and it never occurs in suffixes. Although the vowel /i/ is similarly rare in non-initial syllables, it occurs in suffixes, especially in verbal ones.  

Hill Mari

 Rounded vowels are prohibited in non-inital syllables.

Meadow Mari

Rounded vowels are practically prohibited in non-initial syllables, [ø] and [o] occur in word-final position due to rounding (and F/B) harmony as allophonic alternants of /ə̆/.  The alternation [e] : [ø] occurs in the same position. However, phonemic /e/ does not alternate due to VH.

Udmurt

All rounded vowels but /o/ are prohibited in non-initial syllables (/o/ is quite frequent even in suffixes).  

Hungarian rounding harmony

only short mid rounded vowels alternate with /ɛ/. High and long vowels do not undergo rounding harmony, although any of them can occur in non-initial syllables.

Southern Khanty

Rounded vowels are prohibited in non-initial syllables.

Vakh/Vasyugan Khanty

Although practically all rounded vowels are prohibited in non-initial syllables, alternations /o/ : /ø/  and /ø̆/ : /ŏ/ occur in one suffix, respectively. However, because in both suffixes the alternating vowels are followed by /ɣ/, it is questionable whether these are the rounded vowel phonemes or just rounded allophones of non-rounded ones. The latter assumption is especially problematic in the case of  the /o/ : /ø/ alternation, since full /e/ has no back counterpart. Moreover, non-alternating /u/ occurs in one suffix, but /y/ never. (Cf. complete alternations.)

Nganasan

There could be an /e/ : /o/ alternation due to quasi-rounding harmony and /e/ : /ɘ/ alternation due to F/B harmony, but  /e/ and /o/ are practically prohibited in non-initial syllables and they never occur in suffixes.

Kamas

Rounded vowel phonemes are practically prohibited in non-initial syllables (or at least in suffixes), although rounded allophones of the reduced vowel /ə̆/ (= /ĭ/ and/or /ɑ̆/) occur due to full harmony.