This question was originally inspired by a "Buddhist to Christian" testimony (courtesy of this answer). Specifically, this part of the testimony:
[...] I met her while visiting my best friend, Zlatko. She listened sympathetically to the problems and struggles in my life, emphasising that ONLY GOD could help remove the burdens from my shoulders. I could not work for my “”Salvation””. As I listened I felt different, warmth had crept into my heart, and finally I said a short prayer to welcome the LORD Jesus Christ into my heart as my Saviour. I didn’t know it then, but found out later, that Jovanka my friend’s wife had prayed intensely in “”tongues”” while I was being witnessed to. God later told her the following: “”Marlies, you think it was your good witnessing skills that led Souk to Jesus! No, when Jovanka prayed in tongues she bound the demon sitting on his head that had stopped him from turning to the Lord Jesus Christ.””
When the prayer was over I felt totally at peace and relieved. I was not even afraid of what my parents might say that one of their children had changed from a religion that had been in the family for centuries. In fact, within a few weeks I told them that no longer did I follow the Buddhist religion, but had become a Christian.
It turns out this is not an isolated case. After some searching I was able to confirm that other Christians share similar beliefs regarding the usefulness of tongues in the context of spiritual warfare and deliverance/exorcism. For example:
From https://godencounters.com/praying-gift-tounges/ :
Tongues of war – There is a place where the Holy Spirit goes on the offensive. He battles through us. Sometimes there is a coupling together of the gift of faith with tongues. A rising within occurs and a push or urge to launch out against the demonic forces happens. Sometimes it is the wedding between discerning of spirits and tongues. You combat or attack the dark force, commanding it to flee.
From http://www.demonbuster.com/tongues.html :
During some of our DELIVERANCE sessions when the demon is being stubborn, we pray out loud in TONGUES. Usually, the person immediately covers their ears with their hands because the demons apparently can understand. The demons absolutely hate for you to pray in TONGUES at any time.
By the way, the demons can also speak in TONGUES, but it is coming from the spirit of the devil. You almost cannot tell the difference, which is why we have cast OUT many demons of FALSE TONGUES.
Even Tim Conway, from I'll Be Honest (same ministry as Paul Washer), in the video Has the gift of speaking in tongues ceased? - Tim Conway, referenced an anecdote from the book From Witchcraft to Christ (link) that involved tongue-speaking for deliverance. At 21:09 Tim said:
But the fact is, the gift of tongues was a legitimate, and I believe, is a legitimate spiritual gift, and if somebody has an ability - look, I read a book not too long ago, called 'From Witchcraft to Christ', and the man who was heavily involved in delivering this woman from all these demons that were in her --she was involved in witchcraft, she was involved in Satan's worship--- and a friend of Martyn Lloyd-Jones, he believed he had the gift of tongues of angels, and you know what the woman said, who had been delivered of all these demons? She said, whether he did or not, I can't say but she said "I know this, when he would talk in that language, the demons inside me were terrified and they understood him". So, again, I'm not going to say what God can or can't do. And this was a very godly man, and like I said, he was a friend of Martyn Lloyd-Jones, and it seemed like God had given him special and unique gifts to be able to deal ... Is it possible man can speak in the tongues of angels? Hey, I haven't had it, I'm not going to say first hand. Paul definitely alludes to things like that [...]
Question: What is the biblical basis for speaking in tongues for deliverance and spiritual warfare?