Each Wikipedia article discusses the possibility of the dish/word being related to French pot-au-feu but the articles do not even mention each other.
In looking for other sources you might find the Lao and Vietnamese dishes/words discussed but with no mention of the French.
It seems it is difficult to get to the hard etymological facts, or at least solid likely theories due to such things as folk etymology and nationalism. (The Wikipedia talk page for the Vietnamese dish has some reasonable debate while the one for the Lao dish is very unreasonable and nationalist with a mixture of some logical points, some bad linguistics, and some arguments I suspect to be based on misunderstand but don't know enough.)
The Chinese word
粉 may also play a part, via Cantonese.
So what do we actually know about any relationship between these three words?
All of the languages in question have at least one vowel which might be described as being "like a schwa". In the case of Lao and Vietnamese there seem to be many interpretations for how to map the sounds in these words into IPA. Here are all the variants I have found for the words in question, including comments from Wikipedia talk pages about the tones:
- "the Lao word, feu, uses a neutral tone" (Wikipedia talk page for the Lao word)
/fǝ̌ǝ/(found by Googling)
/fɤˆː/(found by Googling)
- "the Vietnamese word, pho, uses a rising tone" (Wikipedia talk page for the Lao word)
/fə̃ː/(found by Googling)
/fɤ̌/(found by Googling)
/fə ̉ː/(found by Googling)