The properties of the Elixir of Life are said to make one immortal:
The ancient study of alchemy is concerned with
making the Philosopher’s Stone, a legendary
substance with astonishing powers. The Stone
will transform any metal into pure gold. It also
produces the Elixir of Life, which will make
the drinker immortal.
Philosopher's Stone - page 161 - Bloomsbury - chapter 13, Nicolas Flamel
In Philosopher's Stone, Voldemort says about the Elixir of Life:
‘See what I have become?’ the face said. ‘Mere shadow and vapour ... I have form only when I can share another’s body ... but there have always been those willing to let me into their hearts and minds ... Unicorn blood has strengthened me, these past weeks ... you saw faithful Quirrell drinking it for me in the Forest ... and once I have the Elixir of Life, I will be able to create a body of my own ... Now ... why don’t you give me that Stone in your pocket?’
Philosopher's Stone - page 213 - Bloomsbury - chapter 17, The Man with Two Faces
Also in Philosopher's Stone it says the Elixir of Life creates immortality (if drunk regularly, not just one time). How would Voldemort, whose spirit was encased in Professor's Quirrell's body, build a body from the Elixir of Life? It's Quirrell who would have been drinking the elixir anyway, not Voldemort. Is the automatic provision of a physical form implied by the opportunity for immortality through the regular dosing of the Elixir of Life?
Added for clarity: As I commented to NominSim, I'm not asking about Goblet of Fire. I'm asking about what would have happened physically and/or mentally to Professor Quirrell if he had drunk the Elixir of Life in Philosopher's Stone on Voldemort's behalf. How would Voldemort have physically evolved from a spirit to a man while encased inside Quirrell's body? What would have happened to Quirrell? It's already stated in the original question that Voldemort could achieve a new body through the Elixir of Life. I cannot find a suggestion in canon that indicates the Elixir of Life is a potions ingredient. Canon suggests that the Elixir of Life by itself can ensure immortality as long as the witch/wizard in question drinks it as required/regularly.
I don't prefer HP Wikia/Wikipedia answers. I'm looking for a canon-based answer or an answer within the spirit of canon.