There is a negative mitzvah in the Torah not to subtract from the mitzvot of the Torah as it says in the pasuk "אֵת כָּל־הַדָּבָר אֲשֶׁר אָנֹכִי מְצַוֶּה אֶתְכֶם אֹתוֹ תִשְׁמְרוּ לַעֲשׂוֹת לֹא־תֹסֵף עָלָיו וְלֹא תִגְרַע מִמֶּנּוּ".
Nature of the Prohibition
Completely Missing a Mitzvah
- Some understand this mitzvah to mean that if a person fails to fulfill any mitzvah, in addition to violating the actual mitzvah, he also violates Bal Tigrah.
- Accordingly, some ask that every time when a person doesn't fulfill a mitzvah he positively violated Bal Tigra there should be Malkot (lashes) for that?
Uprooting a Mitzvah
- Others argue that it is only if a person has intent to uproot the mitzvah completely.
Fulfilling the Mitzvah in an Incomplete Manner
- Some understand this mitzvah to mean that it is prohibited to fulfill a mitzvah in detracted manner. 
Doesn't Apply When There Is a Rabbanic Enactment
- Some rishonim hold that there's no prohibition of bal tigra if there is a rabbinic enactment to do or not do a certain action. Others disagree.
- Devarim 13:1, Rambam (Sefer HaMtizvot n. 314), Sefer HaChinuch 455
- Turei Even (Rosh Hashana 16b) in explanation of the Rashba. Knesset Rishonim Zevachim 80a fnt. 44* notes that the Ramban Dvarim 4:2 who says that Bal Tosif applies to adding a mitzvah might also agree that neglecting a mitzvah entirely is considered bal tigra.
- Turei Even Rosh Hashana 16a asks on the Rashba that if neglecting a mitzvah is Bal Tigra why isn't there malkot for neglecting a mitzvah and we never found that to be the case.
- Turei Even Rosh Hashana 16a says that according to the opinion that there's no malkot for a lav without a maaseh (Gemara Macot 16a) in a case where one neglected the mitzvah one was passive.
- The Sfat Emet Rosh Hashana 28b writes that the prohibition of Bal Tigra is a general prohibition (lav shebeklalot) and doesn't deserve malkot. He suggests that the same is true of Bal Tosif. Even though the Gemara Eruvin 100a states that there is malkot for violating Bal Tosif that is only malkot mardut, rabbinic malkot. Bet Halevi 1:42 argues that even though Bal Tigra is a lav shebeklalot, Bal Tosif is a specific prohibition and not lav shebeklalot.
- Bet Halevi 1:42 advances the claim that perhaps that whenever a person neglects to fulfill a mitzvah it is considered a violation of Bal Tigra as a kal vechomer. If Bal Tigra applies to doing a mitzvah in an incomplete fashion then certainly it applies to neglecting the mitzvah altogether. However, since that explanation is based on a kal vechomer there is no malkot for that (Macot 5b ein mazhirin min hadin).
- Sfat Emet (Rosh Hashana 28b)
- Turei Even (Rosh Hashana 16b) in his own opinion, Bet HaLevi 1:42
- Sifrei Devarim 13:1
- Rav Hershel Schachter in Ginat Egoz 2:4
- Rashba Rosh Hashana 16a writes that there's no prohibition of bal tosif when the rabbis enacted to do a certain action. His proof is that we keep Yom Tov Sheni and that isn't considered bal tosif since the rabbis enacted it for a reason. Then he extends this argument to bal tigra. The reason that it is permitted not to blow the shofar on the first day of Rosh Hashana when it falls out on Shabbat is because it is considered a rabbinic enactment for a need. Ritva Rosh Hashana 28b agrees.
- The Turei Even Rosh Hashana 16a disagrees because there is no reason a rabbinic enactment could undo a Biblical prohibition of bal tigra. The Haghot Imrei Baruch answers that bal tosif or bal tigra are relevant when the offender is changing the Torah, however, when the intent is to follow the rabbinic enactments there is no change of the Torah. Additionally, the Turei Even asks how the Rashba could explain the Gemara Zevachim 80a that there is bal tosif and bal tigra when dealing with a mixture of blood even though there is a real need and the rabbis should have enacted a solution. Lastly, Turei Even points out that Rashi Eruvin 96a s.v. vod clearly seems to disagree with the Rashba as he asks the question of how we could do Yom Tov Sheni and not violate Bal Tosif and doesn't answer it like the Rashba did.