The standard answer is that all placoderms died by the end of the Devonian, in other words no later than 359 million years ago (MYA). It turns out this is (alas) pretty definitive.
One would think the placoderms were well-prepared to survive. They existed all over the world, in 334 named genera and no one knows how many species, and had evolved not only armor plate and terrifying choppers but also (in some species) developed claspers and modern intromittent sex, which took much of the randomness out of reproduction. (They also, in some species, invented real teeth.) Dunkleosteus terrelli was the king of the placoderms, and nothing in its time could mount any significant challenge to an adult of 7 to 8 meters. (For cool information, see my page here.)
From the author's collection of my favorite placoderm, Dunkleosteus terrelli.
But one might say that if extinction could happen to T. rex, it could happen to anyone, and the Devonian ocean ecosystem crumbled under a double extinction event (more accurately, two events separated by millions of years but close in geologic time.) The first blow was the Kellwasser event. This came between the penultimate phase of the Devonian, the Frasnian, and the last phase, the Famennian. It kicked off about 375-374MYA (just as the famed Tiktaalik rosea was adapting to life out of water) and, according to Dr. Lauren Cole Sallan’s paper (see it here, cited below), caused “spectacular losses in marine diversity involving ∼13–40% of families and ∼50–60% of genera.” It hammered such globally successful groups as trilobites, ammonites, and reef-building invertebrates, making a mess of the oceanic food web and to some degree making the existence of all marine species more precarious. Just as D. terrelli (genus Dunkleosteus, family Dunkleosteidae, order Arthrodira, class Placodermi) and its kin were congratulating themselves for dodging this train of destruction, they met the ecological bulldozer called the Hangenberg event, 359MYA. This calamity wiped out the placoderms for good and left the oceans open for the rise of two now-dominant lineages, the ray-finned bony fishes of the class Actinopterygii and the sharks and rays of the class Chondrichthyes. (The sarcopterygians (coelacanths, lungfishes) and the agnathans (lampheys and hagfishes – the Greek name means “disgusting as hell,” or at least it should) also snuck through.) See NOTE below for a little more information on these extinctions.
There are, in sources like Brittanica.com, claims that one or two species of placoderms escaped the bottleneck of end-Devonian times and made it into the Carboniferous, though they didn’t last long. This is now considered highly suspect at best and very likely false. Raising the question sparked a good discussion in the Devonian Period FaceBook group, and the scientists who weighed in, including Dr. Sallan and Dr. Andrew Bartholomew, are quite certain the claims of placoderms from above the black shale layer which ends where the Devonian ends were based on material that was misidentified or reworked. (I wonder if it was more the latter than the former: it’s hard to imagine chunks of placoderm armor in any strata not being identified, either at the time of collection or in later reviews.) Dr. John Marshall, a Professor at the University of Southampton, pinpointed the sites as lying on the Greenland-Scotland Ridge, with one site in each of the nominate landforms, and posted that he and others had looked at the areas involved without finding any evidence of placoderms.
So that was it for the placoderms. Having crept innocuously into the record in the late Siluran period, introduced long-lasting evolutionary concepts and innovations (although the sharks apparently lost their stashes of placoderm porn and had to invent clasper-based sex all over again), and ruled most of the 60-million-year Devonian, they vanished from the stage, gone but not forgotten.
Numerous sources including:
Sallan, Lauren, and Coates, Michael (June 2010). "End-Devonian extinction and a bottleneck in the early evolution of modern jawed vertebrates," Proceedings of the National Academy of Sciences. 107 (22): 10131–10135. doi:10.1073/pnas.0914000107. PMC 2890420. PMID 20479258.
Balter,Vincent, et al. (2008). “Record of climate-driven morphological changes in 376 Ma Devonian fossils,” Geology, 36(11):907.
Maisey, John G. and Maisey, John. Discovering Fossil Fishes : Your Guide to the Wonders of Prehistoric Ocean Life (1996). Henry Holt.
NOTE: The Kellwasser event (actually a series of events with a variety of possible causes including vulcanism and extraterrestrial impacts) opened about 375 MYA and, by some reckonings, lasted almost up to the Hangenberg event, 359MYA. The latter planet-wide ecological shift, possibly caused by the combination of falling sea levels and the effects on atmospheric and oceanic chemistry from the burgeoning success of terrestrial plant life, changed the oceans so drastically it brought the Age of Fishes to a close and drove into extinction 97 percent of known vertebrate species (remember, these were almost all fishes) with it.