Prohibition on teaching
Tertullian does indeed forbid Christians from pursuing the profession of a school teacher, in De Idololatria, Chapter 10:
Undoubtedly Christians are not allowed to be schoolmasters or teachers since such men are involved in a variety of idolatrous practices.
He saw the profession of his time and culture to be intimately connected with idolatry, in two main ways, the first being that the pagan gods were an essential subject:
they have to make known the gods, their names, genealogies, etc. which, of course, contributes to idolatry
He goes so far as to call the teacher "a catechist of the idols," as in his teaching "he confirms the preaching of the heathen gods contained in literature; he calls them 'gods', which the Bible forbids."
The second objection Tertullian has to Christians working as teachers is that they are forced to participate in pagan festivals
they have to observe the heathen festivals for the sake of their incomes
Apparently it was financially necessary for teachers to participate in the Quinquatria, a festival to Minerva, because there not only did they supplement their regular incomes with tips, they also attracted new pupils. And in order to participate in it, they needed to carry a "tablet with the names of the seven idols" (that is, the seven planets).
Not only this, says Tertullian, but some income had to be dedicated to pagan gods, and in honor of another, the school had to be decorated: "the whole pomp of the devil has to be assisted by [the teacher]."
So we see that, to Tertullian, the profession of teaching was so intimately connected with idolatry that he saw no way for a teacher to avoid sinning. But, as we'll see shortly, he saw great value in education generally, so this prohibition ultimately rests in the sinful activities associated with the profession, not the profession itself.
Studying under a teacher
Tertullian summarizes the natural objection to his view on the teaching profession as follows:
One may object: 'If teaching is not allowed to a Christian, then learning is not so either. This means denying him admittance not only to human education but also to the studia divina'
Tertullian's appreciation for education is evident in his response, in which he argues that "a Christian may go to school":
We must, indeed, admit that a Christian cannot do without literary instruction
So what's the difference, then between being a teacher and being a pupil? Tertullian explains:
If a Christian teaches, he confirms the preaching of the heathen [...] As a pupil, however, a Christian will behave like the man who receives a cup with poisoned contents from somebody not aware of this fact; he will accept it, but not drink from it.
Tertullian clearly saw the importance of literary instruction, and affirmed the importance of Christians studying, even under those who teach false things. However, the close association between the teaching of his day and the honoring and promotion of false gods led him to regard the teaching profession as unsuitable for Christians.
Quotes come from Waszink and Winden's critical edition of De Idololatria, 179ff., which also includes helpful context for Tertullian's argument. An older translation is also available in Schaff's Ante-Nicene Fathers, Volume 3.
The original Latin text, with critical apparatus, is available on tertullian.org.