First off, I apologize for such a general question. I can't think of the name of the tribe and can't remember where exactly I read about them. I just remember I was studying for an exam when I came across this piece of information and was genuinely intrigued. Any thoughts/answers would be much appreciated, this has been bugging me for a while.
I don't know of any middle eastern tribe with such a tradition. However, I know of an old tradition in parts of India, which was similar but not identical with what you describe.
Among the Rajputs (Hindu kings in western India) there was a custom of "Jauhar" in the period of the Islamic conquests, around the 13th - 16th centuries. It was not done routinely when the men went off to war -- it was only done in desperate circumstances when, for example, a fort had been besieged and no aid was forthcoming. Facing certain defeat, the women would build a large bonfire and walk into it, and the men would march out to die in battle. The women's self-immolation was "jauhar", the men's suicide-via-battle was "saka". The two went together.
This was done to avoid dishonor. The usual fate of Hindu women captured by Muslim armies was sexual slavery. The Rajputs were a martial people, and the women picked a particularly painful death by burning to establish that it was not done out of cowardice, but to protect their honor.
There are several instances of jauhar recorded at Chittorgarh, Ajmer, Gwalior and Raiseen, all Rajput strongholds. The Chittorgarh fort was besieged and destroyed 3 times - in 1303 AD by Alauddin Khilji, in 1528 by Bahadur Shah, and in 1568 by Akbar, each time leading to a jauhar. Abul Fazal was a historian at Akbar's court, who wrote the Akbarnama, a history of Akbar's reign. His manuscript has this illustration of the 3rd jauhar of Chittorgarh.