1) It seems to me that the Talmid can touch the shoes and afterwards wash his hands. The Mishnah Berurah (OC 4:41) writes that the reason one must do Netilas Yadayim when one removes his shoes is not because of Ru'ach Ra'ah but because of cleanliness. Therefore, I do not see that it would be forbidden to touch shoes when wearing Tefilin as long as one washes his hands afterwards.

2) I found, bs'd, that the Shulchan Aruch (YD 402:2) states that if somebody heard more than 30 days after the passing that a close relative had died, it is sufficient to remove his shoes as a sign of mourning. If he was wearing Tefilin when he heard the bad news, he is not required to take them off, but it is sufficient to take off his shoes. We see that the Shulchan Aruch states that one may remove shoes when wearing Tefilin (after he removes his shoes he must wash his hands).

3) There is another possibility. The Talmid could hold a cloth or Beged when he removes the Rebbi's shoes, so that he does not even require Netilas Yadyaim since his hands did not touch the shoes directly.

4) Tosfos states that an Eved sometimes wears Tefilin but not all the time:

Avrumi, I think that what you wrote that he is wearing Tefilin now since it is obvious that everyone has to wear Tefilin every day, is in fact written, at least approximately, by Tosfos (96a, DH Aval) who cites the Gemara above (28b) that the Eved does not go free just because he puts on Tefilin in front of his master, but it is not usual that an Eved would wear Tefilin all the time. This seems to mean that an Eved also wears Tefilin sometimes but not all the time.

5) Possibly the Talmid only undid the shoelaces:

I had another idea how to answer this question (even though I do not think it is necessary to say this since I think the first answer I gave, bs'd, is adequate, but this answer may also add a facet in Halachah). It is possible that the Talmid undid the shoelaces of his Rebbi without touching the shoes. Some Poskim say that if one does this it is not necessary to wash the hands afterwards, so this should be more clear that it is permitted to do so while wearing Tefilin.

6) A number of Poskim maintain that there is a Ru'ach Ra'ah on shoes:

I have found, bs'd, that the Mor u'Ketzi'ah (by Rav Yaakov Emden zt"l), near the end of Siman 4, writes that once one has worn shoes at least once, they possess Ru'ach Ra'ah, unlike the opinion of the Mishnah Berurah that I cited earlier. I also found that the Shulchan Aruch ha'Rav (128:27) writes that even if one touches clean shoes, he must wash his hands because of the danger of Ru'ach Ra'ah. So my first answer will not work according to this.

So possibly to answer the question from the Gemara -- how is one allowed to have contact with Ru'ach Ra'ah if he is wearing Tefilin -- we would have to give one of the other answers: he only untied the shoelaces without touching the shoes, or he removed the shoes while holding a cloth between his hands and the shoes. This remains Tzarich Iyun.

7) Here is an answer according to the Chasam Sofer:

a) The Chidushei Chasam Sofer here (DH Aval) provides additional insight into the words of Tosfos (DH Aval). He explains what Tosfos is telling us when he writes that the Eved "does not usually wear Tefilin all the time." The Chasam Sofer writes that there is a difference between what a person who wears Tefilin all day long is allowed to do while wearing the Tefilin, and what a person who wears Tefilin only during Davening is allowed to do. He cites the Gemara in Bava Metzia 105b which states that one may not take garbage outside when one is wearing Tefilin on one's head. The Chasam Sofer writes that it is the only act of work that a person who wears Tefilin all day long is not allowed to do when wearing Tefilin. All other forms of work are permitted.

b) It follows that when it comes to untying shoes while wearing Tefilin, it depends on whether he wears Tefilin all day or not. If he wears Tefilin only when saying Shema and Davening, it is considered disrespectful to the Tefilin if he would untie a shoe while wearing them. In contrast, a person who wears Tefilin all the time is allowed to untie a shoe while wearing them, since the only thing considered disrepectful for him is carrying out garbage; untying a shoe is normal behavior for such a person.

c) Now, we conclude that if someone unties a shoe while wearing Tefilin this proves that he is not an Eved. The only person who is allowed to untie a shoe when wearing Tefilin is someone who wears them all day long, but it is not the way of an Eved Kena'ani to wear Tefilin all day long. Therefore, a Talmid who unties shoes for his Rebbi while wearing Tefilin would not be suspected of being an Eved.

d) So the answer to your question, Avrumi, is that even if we say that one is not allowed to touch shoes while wearing Tefilin, that applies only nowadays when we do not wear Tefilin all the time, while someone who wears Tefilin all day long may touch shoes.

8) Sources that suggest that one may touch shoes while wearing Tefilin according to the Ikar ha'Din:

a) The Shulchan Aruch (OC 43:9) writes that it is permitted for a doctor to take an "Avit Shel Mei Raglayim" (bedpan) in his hands while he is wearing Tefilin, but if he is a Ba'al Nefesh he will be stringent. The Mishnah Berurah (#27) writes that the doctor is doing this in order to examine the patient's urine, and he is not required to take off his Tefilin. The Mishnah Berurah adds that the reason why the Shulchan Aruch here mentions a doctor is that the case which the Rashba (who is the source of the Din in the Shulchan Aruch) was asked about involved a doctor, but in fact anyone is allowed to hold an Avit Shel Mei Raglayim when he is wearing Tefilin. (See also Tzitz Eliezer 13:8 and 18:4 in the name of the Levush.)

b) It does not seem that one can argue that the Heter of the Shulchan Aruch applies only to someone who wears Tefilin all day, because the Shulchan Aruch (OC 37:2) writes that the custom nowadays is not to wear the Tefilin all day, but even so he cites the Heter of the Avit.

c) If one may touch the Avit, then it seems logical to say that one may also touch shoes, and in fact the Minchas Pitim, by Rav Meir Arik zt'l on Orach Chayim 43:9 (concerning the Din of the Avit), cites the Shulchan Aruch (YD 442:19, which is based on our Gemara), where he see that if the Talmid is wearing Tefilin he may put on or take off his Rebbi's shoes. Again, one sees from the fact that the Shulchan Aruch cites this Halachah that it also applies nowadays when most people do not wear Tefilin all day, as the Shulchan Aruch that I cited above writes.

d) All of this seems to point to the conclusion that, according to the basic Halachah, one may touch shoes when wearing Tefilin. It is certainly true that it is good to be Machmir on this, as the Shulchan Aruch (OC 43:9) himself writes that a Ba'al Nefesh should be stringent, but we have shown a few sources, bs'd, that seem to imply that according to the Halachah it is permitted.

9) Rav Chaim Kanievsky zt'l was inclined to say that one may tie up one's shoes or scratch one's head while wearing Tefilin:

a) See sefer Da'as Noteh (vol. 3, p. 445, question 840), where Rav Chaim Kanievsky zt'l was asked if one may tie up his shoes or scratch his head when wearing Tefilin. He replied, "Yitachen she'Mutar" -- "it is probable that this is permitted." In note 1128, the Talmidim of Rav Chaim cite the Shulchan Aruch (YD 442:19) that I cited above, which is based on our Gemara, from which we see that one may put on or take off shoes when wearing Tefilin. The Chasam Sofer on our Gemara, which I cited above, is mentioned but it is pointed out that the Chasam Sofer does not appear to be consistent with the Shulchan Aruch, since, in the time of the Shulchan Aruch, the custom was already widespread not to wear Tefilin all day long but he still permits putting on shoes when wearing Tefilin.

b) Note 1128 to Da'as Noteh also cites the Mekor Chaim, by the author of Sefer Chavos Yair, on Shulchan Aruch OC 43:9 (concerning a doctor touching the bedpan), who writes that since it is a good practice to be stringent not to touch a bedpan when wearing Tefilin, one certainly should not kill lice, even with one's right hand, when wearing Tefilin. The Mekor Chaim adds that one should not even touch a louse while wearing Tefilin since the Shulchan Aruch OC 4:18 writes that one must wash one's hands after touching a louse. It seems that the Mekor Chaim maintains that one should not do something while wearing Tefilin which would necessitate washing one's hands. This is not like what I wrote in my very first answer. However, this is only a Chumra, as we will see, bs'd, and not obligatory.

c) The Mekor Chaim concludes that what he wrote does not constitute a prohibition. We see from the Rema OC 97:3 that if one is not in the middle of Shemoneh Esreh and is being bitten by a louse, he may take it with one's hand and throw it off in the synagogue.

d) The conclusion of a lot of Poskim therefore seems to be that it is permitted to touch shoes while wearing Tefilin, but it is praiseworthy to be Machmir and avoid doing this if possible.

Kol Tuv,

Dovid Bloom