Q. Dear Rov, I was wondering, since there are limitations to the learning of Torah during Tisha Beav, is there a mitzva at all to learn Torah in that day?
The difference would be in a case of doubt. If there is a mitzva of learning Torah one should be stringent to comply with this Biblical Mitzva as opposed when not. Besides, one should indeed make an effort to engage in learning permitted material!
A. Shevet Yehuda (Y.D. 384: 4) and Erech Hashulchan maintain that indeed that there no mitzva at all to learn Torah during Tisha Beav.
However, most Poskim disagree. And the common accepted minhag is to recite the blessing for Torah learning in the morning of Tisha Beav. See also Piskei Teshuvos (O.H. 554: 1).
Rabbi A. Bartfeld as revised by Horav Dovid Pam, Horav Aharon Miller and Horav Chanoch Ehrentreu and Horav Kalman Ochs Shlit'a