It should not matter for proteins but will matter for nucleic acids.  A small portion of aqueous phase will go into the phenol.  The point of saturating the phenol is to fill that compartment in the phenol with aqueous media.  The tris is used to buffer phenol to insure that the "aqueous compartment" is at a suitable pH for what you are doing.  Normally, this compartment will be at a pH of greater than 7.8 for nucleic acid extraction to insure that nucleic acids do not enter the phenol which will be a little bit acidic.  I used a variant of this method to extract extremely dilute proteins.  So I discarded the upper aqueous phase where nuclei acids were and kept the lower phase where the proteins are.  Proteins go to the lower phase (phenol).  The subsequent steps I used diethyl ether to remove the phenol.  The protein would then move into the aqueous compartment that was previously n the saturated phenol.

BTW, if this is for nucleic acids I would tell you to by pre-made tris buffered phenol.  Crystalline phenol will require a subsequent distillation prior to use to remove quinones ( oxidation products) which can attack diester bonds.  Re-distillation above 160 C is not something many labs can do nor would they want to do.

If I may ask, what are you going after in the axtraction?

We regularly prepare Tris-saturated phenol from the crystalline one, but the first step after liquefying it at 55 °C is anyway to saturate it and wash it with distilled water. Then water is exchanged with 1 M Tris/HCl pH 7.4 and finally 2x 100 mM Tris/HCl pH 7.4 for DNA or RNA clean up in a 1:1 misture with CHCl3 (but you can use pure acidic phenol, so, just saturated with water, to extract DNA from a total cell lysate in order to obtain only RNA in the acqueous phase, the DNA will precipitate in the interphase). In our experience, we don't really see evident RNA or DNA degradation due to our phenol preparation, it mostly depends on the tissue from which we extract the nucleic acid and the tissue dissection itself.

Honestly phenol is a quite stable molecule, just an acidic alcohol (that's why for DNA clean up you need to buffer it at neutral pH), and mostly reacts by electrophilic aromatic substitution, but there is not much electrophilic in air or water normally. Possibly it could form hydroquinones reacting with N2O, but I wouldn't know how much of it you find in the air, since it is very reactive with O2 (one of the main problems for our ozonosphere is in fact the NO produced by this reactivity, while N2O is a natural metabolic product of NH3 and NO3- processing by bacteria in the ground).

In any case, after evaluating all the comments, if you wish to prepare your buffered phenol by yourself, the procedure is simple enough. Remove the excess water above the saturated phenol phase, add once an equal volume of milliQ water and let the phases separate O/N at room temperature or at 4 °C (lower temperatures enhance phase separation and minimize most chemical reactivity). If the emulsion does not completely disappear (it will appear as a whitish foamy ring above the clear phenol phase) don't worry, just remove as much of the clear top water phase as you can and add an equal volume of 1M Tris/HCl pH 7.4. Mix well and let the phases separate: this time it will go fast (maybe 30 min) and there will be no emulsion left, due to the increased ionic strength of the water phase which favours the separation of the phenolic component. Again remove the top acqueous phase and add an equal volume of 100 mM Tris/HCl pH 7.4. Do this twice to be sure to equilibrate the saturated phenol to 100 mM Tris.

Work under a fume hood, since phenol burns and is toxic in high amounts (although it has a lot of uses in medicine at low doses!). Use appropriate gloves because it also penetrates skin quite efficiently!

thank you very much for your detailed explanation Mr.Pietro Pilo Boyl.

Just a follow-up question to this: is there any difference between Tris-buffered and citrate-buffered phenol, besides pH?

I want to extract LPS from bacterial cells. The protocol asks for Tris or water-saturated phenol but I only have the citrate-buffered one.