Most authorities somehow narrow the scope of the injunction.
- Only with cantillation
According to R. Qafih (commentary to Hilkhot Tefillah 12:9, note 18) there is a tradition that the prohibition only applies to recitation with the cantillation. For this reason he criticises those (Yemenites) who adopted the practice of reciting the Sh'ma with cantillation, as this allows for the possibility of violating this injunction. This opinion is referenced in KolBo (13) as well.
- Only Torah; not Nakh
Tosafot Temurah (14b) suggest that the injunction only includes verses from the Torah.
- Only unfamiliar texts
The Beit Yosef (OH 49) writes that passages included in the liturgy are not included in the injunction, since everyone is very familiar with the texts, and can easily recite them. This opinion is cited by Abudirham (Birkhot HaShahar) as well, who writes as follows:
שכל דבר שרגילין בו ושגור בפי הכל אין בו משום דברים שבכתב אי אתה
This is also the opinion of Rabbenu Tam cited by Mordekhai (Gittin 470). This is also the view presented in Shulhan Arukh (OH 49:1).
- Only a problem for being motsi others
Abudirham (there) cites Rif and Rosh as holding that it is only a problem when being motsi others, but individuals reciting Scripture for themselves is fine. This is suggested by Tosafot (Temurah 14b) as well, and is the opinion of the Yereim (268) and Gra (OH 49). R. Hayyim Zimmerman in Binyan Halakha (introduction to Mishneh Torah p. 2) similarly states that the problem is only in transmitting Torah to a congregation; not in reciting to oneself. He attributes this view to Rif, Rashi, Rambam, and others.
- Only when reading from a Torah scroll
Abudirham (there) cites the Yerushalmi that the problem is only to recite a passage by heart while in the process of reading from the Torah, since people will assume that the parts recited by heart are not found in the text. This Yerushalmi is also quoted by KolBo (13), and by Radvaz in a responsum (4:135). This is the opinion of Rambam as well, according to Adnei Yad HaHazaka to Hilkhot T'fillah (12:8), and others (cited here).
- Only a text which is never obligatory
According to Rabbenu Yonah (Berakhot 5a), citing the opinion of R. Sh'mo Min HaHar, any text whose recitation is ever obligatory, may be recited even when it is not obligatory. [i]
- Only as a form of study; not prayer
According to the Tosafot HaRosh to Yoma (68b) recitation of Hallel and Malkhiyot Zikhronot and Shof'rot is okay since the scriptural verses are just being quoted as a form of prayer and praise.
While not limiting the scope of the injunction the Hikrei Lev (OH 13) writes that it is not an actual injunction, but merely a guide to the proper way to learn Torah. One who recites the Written Law orally does not fulfill the mitsvah of learning Torah.
In a similar vein, the Tosafot HaRosh to Yoma (68b) describes it as a mere mitsva min hamuvhar, as he quotes R. Yitshak of Dampierre in his Tosafot to Sotah (40b).
[i] He does not say exactly what sort of obligation renders a text permissible to be recited, to clarify why it is ever forbidden to recite passages from the Torah, if the whole Torah must be read.