The Shulhan Aruch (O.H. 47:2) states that Birkot Hatorah are recited whether one will study Tanach, Mishna, Gemara, and according to the Rama (ibid.), Midrash. Indeed, when one recites Birkot Hatorah in the morning, one immediately recites verses from the Torah so that the blessings take effect on some sort of Torah study and that they not be recited in vain.
Since the recital of verses from the Tanach is included in the precept of Torah study, there is a question as to whether or not they may be recited prior to reciting Birkot Hatorah. The Shulhan Aruch (O.H. 46:9) says that one should not recite verses before one says Birkot Hatorah even if they are recited in the form of supplications, and that even though there is a lenient opinion, it is preferable to take the strict approach. This comes into play, for example, during the period of Selihot, which contain many portions from the Tanach and which are often said at dawn and possibly before one recites Birkot Hatorah. The Rama (ibid.) discusses a disagreement among the rabbis as to the permissibility of reciting supplications which contain verses before Birkot Hatorah. Practically speaking one should recite all of Birkot Hashahar including Birkot Hatorah before saying Selihot or before reciting any verses.
Rabbi Ovadia Yosef (Yabia Omer, vol. IV, § 7) states that if one hears Kedusha but has not yet recited Birkot Hatorah, one should respond “Kadosh, Kadosh, Kadosh” and “Baruch Kevod…”, but when it comes time to respond “Yimloch”, one should only ponder it. As well, Rabbi Ben Zion Abba Shaul (Or Lezion, vol. II, ch. 4, § 4) says that if one hears Vaya’avor before Birkot Hatorah, one may respond with “Hashem Hashem E-l Rahum…” since it is recited in the form of a supplication. Verses which are not part of the prayer, however, may not said.
Magen Avot (Orah Haim, § 47) records that there was no clear custom in Morocco as to the permissibility of reciting verses before Birkot Hatorah, since for the most part, people recited all of Birkot Hashahar at home before arriving at the synagogue for the prayer. Some question Magen Avot and point out that in Siddur Tefilat HaHodesh and others, certain supplications are written before Birkot Hatorah. Nevertheless, it is not always accurate to define a custom based on what is written in a Siddur, especially in light of the common practice of reciting Birkot Hashahar and Birkot Hatorah early on in the day at home. Furthermore, none of the known Moroccan rabbis wrote that there is a clear custom to allow the recitation of verses before Birkot Hatorah. With this and the Shulhan Aruch’s position in mind, one should be strict and not recite verses before Birkot Hatorah.
Summary: One should not recite any verses from the Tanach before reciting Birkot Hatorah