What are major languages which declined/extinct during Turkification of Anatolia?

All the answers posted are very good, and a more substantial contribution than I will make. I agree that in all likelihood, by the time the Seljuks came to town, the indigenous Anatolian languages were long gone, and it was all about the retreat of Greek and Armenian. But I was A2A’d.

So I’ll talk about Greek.

What do I know? I’ll draw on the survey in Modern Greek in Asia Minor; a study of the dialects of Siĺli, Cappadocia and Phárasa, with grammar, texts, translations and glossary : Dawkins, R. M. (Richard McGillivray), 1871-1955.

• The collapse of Greek language and Christianity in Western Asia Minor in the 13th century appears to have been quite rapid: a matter of generations.
• Though I haven’t seen solid evidence for this, it seems that the substantial Greek populations in the Western Asia Minor coast date from Ottoman times, with Greeks settling the coast from the nearby islands. The dialects of the coast are certainly close to those of the Aegean islands. Dawkins concurs, speaking of both settlement from the islands, and a wave of migration out of Greece in the 18th century.
• We know that Bithynia was resettled by both Greeks and Bulgarians in the 1500s–1600s. In fact, there was even a Tsakonian colony on the mouth of the Gönen river, which probably dates from the 1700s.
• I think the Greek population in European Turkey was continuous.
• The Greek population in the Pontus was continuous, and if anything expanded, with mining colonies reaching far into the Black Sea hinterland: Ak Dağ, Buğa Maden, Bereketli Maden, Nevşehir, Ürgüp, Keban Maden, and around Şebinkarahisar.
• There were two distinct regions where Cappadocian Greeks lived: 6 villages around Pharasa, and 20 villages in Western Cappadocia. The language had substantially retreated by the time Dawkins surveyed them in situ; most Christians in the region already spoke Turkish, and particularly in South Western Cappadocia (e.g. Ulağaç), the Greek spoken was heavily influenced by Turkish.
• There was isolated Greek-speaking communities in Livisi (Kayaköy near Fethiye), the town of Sille near Konya, and a dialect that had died out by 1900 in Gölde of Lydia (near Kula, Manisa).

So, tl;dr:

Greek was vibrant in the Pontus; retreating in Cappadocia (and anecdotally the other remaining old settlements as well, with the possible exception of Livisi); and wiped out everywhere else in Anatolia though Turkification; the substantial Greek-speaking population in Western Asia Minor was the result of later resettlement.

I’m pretty sure this map from Wikipedia (File:Anatolian Greek dialects.png) is overstated for both Cappadocian and Mainstream Greek, but it’s a start: