Does pronunciation of a vowel change with tones?
First, it is important to distinguish between the phonemic pronunciation and the phonetic pronunciation. The first represents how native speakers interpret sounds, the latter interpret how the sounds are effectively performed.

The Pinyin final -iu is phonemically pronounced /jəu/, not  /iou/ (reference: https://en.m.wikipedia.org/wiki/Standard_Chinese_phonology#Allophones). When combined with the Pinyin initial l-, that Pinyin final becomes the Pinyin syllable liu, phonemically pronounced /liəu/. 

Anyway, much more important to a language learner is the phonetic pronunciation. The phonetic pronunciations of -iu may be [jəu], [jou] or even [jo̝u] (vowel O raises due to assimilation to the adjacent high sounds /j/ and /u/), which sounds similar to /ju/. The latter is the pronunciation in the audio recordings of the Pinyin syllable "liu" in the provided Pinyin table. The same also occurs in the audio recordings of the other Pinyin syllables with the final -iu.

Regarding the 3rd tone pronunciation of a Pinyin syllable with the final "-iu", when pronounced in isolation, it falls ([jo]) and then rises ([u]); in this way, this tone has an effect of separation and therefore tends to block the vowel assimilation that I mentioned before. That's why the [o] sound is heard in this tone and barely heard in the other tones in the audio recordings of the provided Pinyin table.