Many times I have heard elders saying that lineage must be continued so that Pitris/ancestors can born again as human. Are there any scriptural references about it?
Sons are required because if they keep performing the prescribed religious rites like Tarpanas etc on earth, then that enhances the heavenly life of the departed ancestors. It is not needed for their rebirths like you have said.
You'll understand once you read the following passage:
From Vashishta Dharma Sutras 17.13:
‘A debt he pays in him and immortality he gains, the father who sees the face of his son born and alive’ (AB 7.13).
‘Eternal are the worlds of those men who have sons. A sonless man has no world’––so states a vedic text (cf. AB 7.13).
And there is the curse: ‘May our enemies be childless!’ (RV 1.21.5). There is also the vedic saying: ‘Through offspring, O Fire, may we attain immortality’ (RV 5.4.10). Through a son one gains the worlds; through a grandson one attains eternal life; and through the son’s grandson one gains the crest of the sun.
(Here RV stands for Rig Veda and AB for Aitreya Brahmana).
You'll get more clarifications from the following passage which is from Apasthambha Dharma Sutras 2.24.8:
The scriptures declare, moreover, that immortality consists of offspring: ‘In your offspring
you are born again. That, O mortal, is your immortality’ (TB 126.96.36.199). Furthermore, we can see with our very eyes
that the son is a distinct clone of the father himself. One can even see that they are identical, only the bodies are
distinct. And the sons, as they continue to perform the prescribed rites, increase the fame and heavenly life of their departed ancestors. Each subsequent generation does the same for those that preceded it. They dwell in heaven until the dissolution of creation. At the new creation, they serve as the seed’, says the Bhavisyat Purana. And there is also the declaration of Prajapati:
Study of the triple Veda, studentship, procreation, faith, austerity, sacrifice, giving gifts––those who perform these dwell with us. Anyone who praises other things becomes dust and perishes.
(Here TB refers to the Taittiriya Brahmana)
So, for one's own immortality concerns one needs offsprings to start with. Furthermore, it is also needed so as to make the deceased ancestors live happily in their own realms for ever till dissolution.
The 2nd quote/passage clearly rules out the possibility that continuing the lineage is to be done for the purpose of the ancestors' rebirth as humans on earth (because they are stated to live in their own realms till dissolution).
But, if you're talking about the father being born again as the son, only then it's seemingly true.