The most endangered language of the region is certainly Northeastern Neo-Aramaic (NENA), or Sûreth (ܣܘܪܝܬ). Once, Aramaic was the lingua franca of the Middle East, spoken in various dialects from Gaza to Basra, and written as far afield as China and India. By the early 20th century, it was restricted to a few hundred far-flung mountain villages; the largest dialect group, NENA, was centered on the Christian (Assyrian and Chaldean) villages of the Mosul Plain, such as Tel Kef (Telkepe) and Qaraqosh, and across the border in Iran and Turkey; a detailed map is available at Cambridge's NENA Database. Today, those who have stayed behind in ever harder conditions are substantially outnumbered by their diaspora in cities such as Detroit or Sydney, whose children increasingly just speak English - and, as of the past couple of days, media accounts suggest that fleeing refugees have left the Mosul Plain villages practically empty. Their exodus is rather reminiscent of what happened about a century ago: during the Armenian/Assyrian Genocide, the NENA-speaking Assyrians of Hakkari fled from Turkey never to return, taking refuge in Iraq and finally in Syria. It remains to be seen whether this exile will be as lasting as the previous one. If you're wondering how the language sounds, the NENA Database site has a number of recordings, some transcribed, such as The Story of the Cobbler; others can be heard at Semitisches Tonarchiv.
While Kurds prefer to consider Kurdish as one language, the two main Kurdish varieties of northern Iraq - Sorani and Kurmanji - are strikingly different from one another, and are usually considered as separate languages by academics. The smaller Gurani language, (see DOBES), spoken in northwestern Iraq and also commonly labelled Kurdish, doesn't even belong to the same branch of Iranian as Sorani and Kurmanji. Many of its speakers belong to loosely Shia-affiliated minority religions, such as the Ahl-i Haqq and the Shabak, considered by ISIS as beyond the pale.
The other minority group unfortunate enough to have been pitched into the headlines, Yezidis, do not have a language of their own; they speak Kurmanji Kurdish. However, the Yezidis are associated with a unique writing system. In the early 20th century, manuscripts summarising Yezidi beliefs written in a unique alphabet (such as the Meshefa Resh "Black Scripture") came into the possession of Western researchers, and the alphabet in question duly found its way into compendia such as Diringer (1968). Later research, though, suggests that both these manuscripts and the alphabet they were written in were created for Western consumption, likely by a non-Yezidi bookseller, rather than representing a Yezidi tradition (Kreyenbrook and Rashow 2005, EI).
The region's Turkmen, many of whom have also apparently been persecuted by ISIS for their Shiism, speak a Turkic variety close to Turkish and Azeri. From what little information I've seen, it seems unlikely to qualify as a separate language, but does not seem to have attracted much research.
The Arabic dialects of northern Iraq - the so-called qeltu dialects, for their unique pronunciation of the word "I said" - are also quite interesting in their own right; the spoken Arabic dialect of Abbasid Baghdad seems likely to have belonged to this group. However, that is another story for another day...