The Gemara in Shevuot 29a establishes that an oath taken in vain — defined there as swearing to contradict what is publicly known, to refrain from testifying, or to annul a mitzva — carries biblical liability for intentional violation, and the Rambam in Mishneh Torah, Blessings 1:15 draws a direct analogy from this, ruling that one who recites an unnecessary brachah 'bears the divine Name in vain and is like one who swears in vain,' framing the prohibition in the language of the biblical verse lo tissa.
The core machlokes is whether this equation with lo tissa constitutes a genuine biblical prohibition or merely a rabbinic one supported by an asmachta: Magen Avraham 215:6 reports the view of Tosafot and the Rosh that the prohibition is only mid'rabbanan — the verse being a mere asmachta — while the Rambam's position, he notes, is that it is a full biblical violation, and he observes that the plain reading of the Gemara in Temura likewise supports the d'oraita view.
Mishnah Berurah 215:20 sharpens the distinction: when the divine Name is pronounced within a brachah in a manner of praise and thanksgiving, the preponderant view of the Rishonim is that the prohibition is rabbinic and the verse an asmachta; but when one utters the divine Name in vain outside of any brachah altogether, it is agreed by all to be a biblical violation, derived from the positive commandment of 'you shall fear the Lord your God,' which the Gemara in Temura reads as a prohibition against pronouncing the Name unnecessarily.
Arukh HaShulchan, Orach Chaim 215:5 follows the Tosafot-Rosh position that a brachah l'vatala is technically not a full biblical lav in the same way as swearing falsely or uttering the Name without a brachah, and that the Gemara's language of 'transgressing lo tissa' is an asmachta — yet he stresses emphatically that the prohibition is nonetheless an extremely grave one.
Haamek Davar on Shemot 20:7 reads the prohibition of lo tissa broadly, noting that Chazal derived from it several distinct applications, including one who recites an unnecessary brachah, and that in every such case the verse's conclusion — 'the Lord will not hold guiltless' — applies, even if the specific consequences differ according to the gravity of each particular instance.
Yitzchak Yeranen on Mishneh Torah, Foreign Worship 5:11 concludes, on our reading of the cited page, that pronouncing the divine Name in vain — including in a brachah l'vatala — is a biblical prohibition according to the Rambam and the majority of the greatest authorities, reasoning from the parallel between the forbidden utterance of the names of idols and the forbidden utterance of the divine Name without cause.