The datation of Dharmakīrti is a topic I am not competent enough to speak about, but I will nonetheless try to summarise other people’s arguments.
The departing point is the traditionally accepted date of Dharmakīrti, namely 600–660, settled by Erich Frauwallner mainly on the basis of the reports of Chinese pilgrims, and especially on the fact that Xuanzang did not mention Dharmakīrti. Recently, Helmut Krasser suggested to reconsider the evidences. Xuanzang’s silence, he meant, is not an evidence at all, since it might have well been the case that Dharmakīrti was deliberately ignored in the curricula at Nalanda because of his critical attitude towards the Buddha’s word (which is not in itself an instrument of knowledge according to Dharmakīrti) and that this is the only reason why Xuanzang did not hear about him. As for the pars construens, Krasser suggests that Dharmakīrti must have influenced Bhāviveka (and not the other way round) and that this influence is recognizable in several topics, e.g., in the increased importance of the topic of omniscience and in the so-called sattvānumāna. Thus, Dharmakīrti must have lived well before the commonly acknowledged date.

During Vincent Eltschinger’s Habilitation’s Defence and now at the IABS the topic has been resumed by Eli Franco (Birgit Kellner, Patrick McAllister, Ernst Steinkellner and others have also taken part to the discussion). Franco’s pars destruens regarded the fact that one finds evidences of the Buddha’s omniscience as a distinct philosophical topic already in the Spitzer Manuscript and that the topic has, thus, not been introduced by Dharmakīrti in the Indian debate. As for the sattvānumāna, Franco contends that its formulation by Dharmakīrti and by Bhāviveka is completeley different and that the only common element, namely sattva as a probans is already found in Uddyotakara. McAllister has suggested during the discussion and “on behalf of Helmut” that the inferences are not that different. They are applied to a different locus (pakṣa), but their probans (namely, anityatva in Bhāviveka and kṣaṇikatva in Dharmakīrti) could be the same. In fact, after Dharmakīrti anityatva is considered to be the same as kṣaṇikatva, so that if Bhāviveka lived before Dharmakīrti, the inferences would be different, whereas if he lived after him, they would be equivalent.
Long story short, according to Franco, the similarities between Bhaviveka and Dharmakīrti are far too vague and general: they could have been “ideas in the air”, rather than precise references. And even if there were direct correspondences, there can still have been a third source. We cannot assume that what we have was actually all that there was. In this connection, Birgit Kellner has pointed out the fact that there is no reason to believe that what we have is all that was circulating at that time. If I (EF) am allowed to step in, this is in my opinion an important thing to be kept in mind while looking for the source of an innovative element in an author’s thought (as paradigmatically done by Erich Frauwallner in his attempt to reconstruct the history of Indian philospohy). An example is the case of the development of Kumārila’s thought from vyāpti to niyama discussed here, but the general point regards the fact that while looking for a source, we are left with far too few candidates.

As for the pars construens of his argument, Franco stressed the silence by Xuanzang, that by Candrakīrti and then silence of Jain authors. The first Jain who refers to Dharmakīrti is in fact Akalaṅka (720–780). All of that has been explained by Krasser as the result of the suppression of Dharmakīrti’s thought from the curricula at Nalanda. But, Franco noticed, we have no evidence neither of this suppression, nor of the later rediscovery of Dharmakīrti in Nalanda. Ernst Steinkellner observed in this connection that curricula are, even nowadays, slow to be updated and Dharmakīrti’s thought was very complex. It might thus be that he could find his place in the curricula only after a generation of commentaries which had clarified his innovations (incidentally, this “softer” explanation is not what Krasser had thought and claimed to have been the case).

A last point: Moving Dharmakīrti back would have a great impact on the chronology of Indian philosophy. For instance, we now know more or less for sure that Dharmakīrti influenced Kumārila, who was influenced by Dignāga. Establishing independently Kumārila’s date would thus lead one to important conclusions regarding Dharmakīrti’s date, too (and vice versa).

For further thoughts on Bhāviveka and Dharmakīrti’s dates and relation, see this post. For some elements towards a date of Kumārila, see this post. This post is a part of a series on the IABS. For the others, see here. Please remember that these are only my first impressions and that all mistakes are mine and not the speakers’ ones.
Full disclosure: I might be implicitly biased in favour of Helmut Krasser, because he was a friend, my former boss and because he is no longer there to defend himself.