Gemara

88b (Mishnah): If a Chacham says that there are five Parshiyos in Tefilin, to add to what Chachamim explained, he is liable (for Zaken Mamrei).

(R. Elazar): A Zaken Mamrei is liable only if ... it is possible (but forbidden) to add to the law, and adding invalidates it. The only example is Tefilin.

Question: Also regarding Lulav, if one adds other species to the four, he invalidates the Mitzvah!

Answer: If Ein Tzarich Eged (the species need not be tied), holding additional species at the same time has no consequence. If Tzarich Eged, the bundle (with the extra species) was already invalid (from when he tied it.)

Question: If one adds additional Tzitziyos, he invalidates the Mitzvah!

Answer: If the Torah does not require even one knot (after inserting them through the corner), extra strings do not invalidate the Mitzvah. If the Torah requires tying the strings, it was invalid from the start (due to the extra strings)!

Question: If so, also adding to Tefilin does not disqualify a valid Mitzvah! If he made four boxes and put a fifth next to them, it does not invalidate the Mitzvah. If he made five boxes the beginning, it was invalid from the start!

Answer: (Really, he made four, and put a fifth next to them.) R. Zeira taught that if the outer boxes are not exposed to the air, the Mitzvah is invalid.

Temurah 16a (Rav Yehudah): Three thousand Halachos were forgotten during the mourning over Moshe's death. Yehoshua refused to ask Hash-m, for "Lo va'Shamayim Hi." Shmuel refued to ask Hash-m, due to "Eleh ha'Mitzvos" - a Navi may not teach a new (mid'Oraisa) law through prophecy.

Rishonim

Rambam (Hilchos Mamrim 4:3): If a Chacham ruled to make four boxes properly, and add a fifth and stick it outside them, he is liable for Zaken Mamrei. The outermost box must see the air. He would be exempt for rulikng about a Mitzvah such as Lulav or Tzitzis.

Rebuttal (Ra'avad): Also for Lulav or Tzitzis, the addition disqualifies what was there to start! Even if the Zaken Mamrei caused people to transgress Bal Tosif, he is exempt unless he disqualified the Mitzvah.

Rosh (Sukah 3:14): Bal Tosif does not apply to taking extras of the four Minim, even according to the opinion that Lulav Tzarich Eged, unless he adds a different species.

Ramban (Devarim 4:2): Bal Tosif forbids also making a new Mitzvah. Chachamim were debating whether we may be Mechadesh Mikra Megilah, until they found verses for it. The Torah commands Chachamim to make fences, as long as it is known that it is a fence.

Igros Moshe (OC 1:14:1): The Ramban says that a Navi cannot be Mechadesh new laws due to Bal Tosif. The Gemara did not ask about Ner Chanukah, reading the Megilah, or Shaloch Manos. It asked only about the final letters of the Aleph-Beis, Shi'urim and the Halachos forgotten during the mourning over Moshe. We cannot say that Bal Tosif applies to fulfilling a Torah obligation! Rather, the Gemara asked that we should not heed a Navi who teaches a new law through prophecy. The Yerushalmi did not ask that we may not add Mikra Megilah due to Bal Tosif, rather, that a Navi cannot obligate Yisrael. It seems that the Yerushalmi holds like the Bavli, that they could enact, just Chachamim were concerned lest people consider this contrary to Moshe's Nevu'ah "these are the Mitzvos" (and there will be no more). Even if the Yerushalmi held that Mikra Megilah was a problem of Bal Tosif, why does the Ramban rule like it against the Bavli?

Maharal Diskin (Re'eh 13:1 DH Es): If one did something only because he thought it is a Mitzvah, does he transgress Bal Tosif? The Rambam says that if Beis Din ruled to make a new Mitzvah, they transgress Bal Tosif. If the people transgress, Beis Din would also transgress Lifnei Iver for making the people transgress Bal Tosif!

NOTE: It seems that this is unlike the Rashba (below).

Rashba (Rosh Hashanah 16a DH Lamah): Bal Tosif is what a person decrees on his own. An enactment of Chachamim is not Bal Tosif.

Poskim

Shulchan Aruch (OC 11:3): If Tzitzis became untwined and there are 16 ends on a corner, it is Kosher, as long as Kedei Anivah (enough to tie a string onto itself, or around all the strings) remains twined.

Bi'ur Halachah (DH v'Na'asu): If one string became totally untwined and Kedei Anivah did not remain twined, surely the Tzitzis are Kosher, but to avoid Bal Tosif, he should explicitly say that he is unhappy with the extra, or he should cut off the extra to be left with eight.

Chazon Ish (3:16 DH Emnam): Extra strings do not disqualify according to the opinion that the Torah does not require a knot. The Anaf (what hangs straight after the windings) need not be tied, so if it became untwined, it is Kosher, even if he intends to add. If a string became untwined before the knot, but it was intact in the knot, it is not clear if Bal Tosif applies.

Pri Megadim (Petichah Koleles to Shulchan Aruch, 1:35): For anything that must be tied, e.g. Tzitzis, Lulav according to R. Yehudah, and Tefilin (it must be within the leather) one transgresses Bal Tosif even for the same species, e.g. an extra Lulav, Tzitzis, or an extra Parshah of Shema. For adding a different matter, one transgresses even if the Mitzvah need not be tied, e.g. a different Berachah to Birkas Kohanim. When a Mitzvah is not tied, one who repeats it does not transgress, e.g. to say a Berachah of Birkas Kohanim twice.

Levush (cited in Divrei Chamudos 19, on Rosh, Hilchos Tefilin after Menachos): One may wear many garments with Tzitzis on them. Bal Tosif does not apply. I am unsure whether the same applies to Tefilin.

Chasdei David (Korbanos 8 DH v'Adayin p.135): Even if one who wears two pairs of Tefilin transgresses Bal Tosif only if he intends, we forbid mid'Rabanan even Stam, for it looks like he transgresses. The Tur says that one transgresses only for five boxes, why must one intend that the other is mere straps? Also, the Gemara says that one transgresses for two pairs! Rather, since the pairs are not tied to the other, since we hold that Mitzvos need Kavanah, one transgresses only with intent. One must intend that the other pair is straps to permit even mid'Rabanan.

Tosfos (Rosh Hashanah 28b DH u'Mina): One who repeats a Mitzvah mid'Rabanan transgresses Bal Tosif. Similarly, one who takes extra Hadasim and Aravos does not transgress, even according to the opinion that Tzarich Eged.

Minchas Chinuch (454 DH uv'Emes): Why does Tosfos equate repeating a Mitzvah with additional Esrogim? There is a minmal Shi'ur to "Yom Teru'ah", "on the first day take", and "at night, eat Matzos", but one may increase! The words connote that the Mitzvah applies the entire day or night! However, if verses teach how many strings or Parshiyos, perhaps adding is Bal Tosif! Tosfos says that if one already wore Tefilin today, the time of the Mitzvah passed, and he transgresses only with intent. It is a Mitzvah to wear them the entire day! This is why one Tana forbids wearing two pairs due to Bal Tosif. We do not say that after being Yotzei for a moment, he transgresses only with intent.

Shulchan Aruch (OC 34:2): One with Yir'as Shamayim will make Tefilin like Rashi, and also like R. Tam, and wear both. He intends to be Yotzei with the Kosher pair; the other pair is merely straps.

Magen Avraham (3): If one wore two pairs of Kosher Tefilin, he transgresses Bal Tosif (Eruvin 96a, Beis Yosef (651 DH Aval), Magid Mishneh Hilchos Lulav 7:7). The Levush and Divrei Chamudos were unsure about this. They overlooked these sources. There is no proof from Sanhedrin, which discusses one who wrote another Parshah not connected to Tefilin.

Suggestion: The Magen Avraham holds that Sanhedrin 89 must discuss a fifth attached box, in order for R. Zeira's law (if the outer boxes are not exposed to the air, it is Pasul) to apply. If Bal Tosif applies when it is attached, we would not need R. Zeira's law! Rather, since it was initially Kosher, we say that they are independent, and it is not Bal Tosif. The extra box is like a mere strap. However, the Gemara says otherwise! This forced the Magen Avraham to distinguish between adding a new Parshah, and adding a repeat of one of the Parshiyos of Tefilin. In the former case, we say that since it was initially Kosher, the extra box does not disqualify. In the latter case, it disqualifies. Tbis does not depend on whether or not it is attached. Only R. Zeira's law depends on this.

Rejection (Igros Moshe OC 1:14:3): When it is not attached, it should not depend on whether the addition is a Parshah of Tefilin. In this case.the Magen Avraham could admit that even the same Parshah does not disqualify. He said that Sanhedrin discusses another Parshah, for it discusses an attached box. Bal Tosif would apply to a repeated Parshah even if it were not attached, just like it applies to wearing two pairs of Tefilin, for there is no logic to say that one is the Mitzvah and the other is mere straps. When it is attached, the Magen Avraham disqualifies only if a Parshah was repeated. Really, even another Parshah disqualifies. We do not say that the repetition is independent, for we find that a fifth species disqualifies Lulav if it must be attached, even if four were attached and a fifth was added later. The same should apply to Tefilin! Rather, we must say that Tefilin are different, for all the boxes must be from one piece of leather.