Mishna Berura Source Sheet (160:3)

(ג) מים שלפני הנפח אעפ"י שלא נשתנו מראיהן פסולים מפני שבידוע שנעשה בהם מלאכה דהיינו שכיבה בהם הברזל.

ושלפני הספר אם נשתנו מראיהם פסולים ואם לאו כשרים:

(3) Water that is in front of a blacksmith is pusul even if it hasn't changed in appearance, because it's known that he does work with it (that is, he cools metal with it). Water in front of a barber is pusul if it's appearance changed but kosher if it has not.

תוספתא ידים א׳:ו׳

(ו) מים שלפני הנפח אע"פ שלא נשתנו מראיהן אין נוטלין מהן לידים מפני שבידוע שנעשתה מהן מלאכה.

מים שלפני הספר אם נשתנו מראיהן פסולין ואם לאו כשרין.

Tosefta Yadayim 1:6

(6) Water that is in front of a blacksmith -- even though it has not changed in appearance, they may not pour it over their hands because it is known that he has done work with it. Water that is in front of a barber is unfit if it has changed in appearance, and if not it is valid. (See Rash MiShantz there)

אליה רבה על שלחן ערוך אורח חיים ק״ס:ח׳

(ח) [ח] [לבוש] דמספיקא וכו'. אבל הנחלת צבי כתב זה לשונו, ואני אומר שאינו דומה לנפח כיון שאינו אלא שטובל להשרות בו השערות דומיא לטובל ידיו וטח בפניו הגלוסקא דכשר, עד כאן. ולפי זה אף בידוע שטבל ידיו כשר וצריך עיון.

שוב מצאתי בתורת הבית הקצר דף נ"ט להדיא כדברי לבוש, ולנחלת צבי ודאי אשתמיטתיה, וכן משמע בתוספתא שהביא הר"ש פרק קמא דידים:

Eliyah Rabbah on Shulchan Arukh, Orach Chayim 160:8

...

Toras HaBayis is clear that the melacha that is done by a barber does make the water passul, not like the Nachalas Tzvi: Torah HaBayis (Bayis 6, Shaar 2) writes:

In other words, Nachalas Tzvi is learning that the reason the water in front of a barber is okay is because the type of work a barber would do with it isn't the kind of work that will passul it even if we know he used the water.
The reason that the washing a barber does is not considered a melacha according to those opinions is because of the mode of washing the barber does.
In the course of cutting hair, his hands get hair on them and he dips his hands into the tub in front of him. This is very similar to the washing that a baker does where he dips his hands into the water to smear onto the bread. Therefore even if we know the barber washed his hands in the water, it will still be kosher for netilas yadayim.
This is in contrast to the blacksmith who regularly puts the actual tools into the water, and this is certainly using making use (melacha) of the water.

Although there is a possibility that the barber may have used the water to wash off his scissors and other equipment, which would certainly be a melacha like that of a blacksmith, we aren't concerned that he will do that, and since it would only be a safeke if he did, because of the off chance that he may have washed off his utensils, that doesn't passul the water (see seif 14) - MB

The Levush and the Toras HaBayis (as well as the mashmaos of how the Elya Rabba sides) argue and say that the reason why the water in front of the barber is kosher for netilas yadayim is because it is only a safek whether work was down with it.
But if we would know that work was down with it, it would be passul.

Pri Megadim says that even if you were make the case that a barber washing off hair from his hands in water does not constitute a melacha, the practice that current day barbers have which is to wash off soap in the tub before them is certainly considered that melacha was done to the water according to all opinions.