What is the Biblical basis for saying one won't be healed if they lack sufficient faith?

Consider the Tree of Life and the Tree of the Knowledge of Function and Dysfunction

but of the tree of the knowledge of good and evil you shall not eat, for in the day that you eat of it you shall surely die.” (Genesis 2:17 NKJV)

and~from~TREE the~DISCERNMENT FUNCTIONAL and~DYSFUNCTIONAL NOT you~will~EAT FROM~him GIVEN.THAT in~DAY you~EAT FROM~him >~DIE you~will~DIE (Genesis 2:17 Mechanical Translation of the Torah)

You will know what Tree of Thought the person eats from by their fruits.

“Beware of false prophets, who come to you in sheep’s clothing, but inwardly they are ravenous wolves. 16 You will know them by their fruits. Do men gather grapes from thornbushes or figs from thistles? 17 Even so, every good tree bears good fruit, but a bad tree bears bad fruit. 18 A good tree cannot bear bad fruit, nor can a bad tree bear good fruit. 19 Every tree that does not bear good fruit is cut down and thrown into the fire. 20 Therefore by their fruits you will know them. (Matthew 7:15-20 NKJV)

Defining the Fruits of the Tree of Life

But the fruit of the Spirit is love, joy, peace, longsuffering, kindness, goodness, faithfulness, 23 gentleness, self-control. Against such there is no law. (Galatians 5:22-23 NKJV)

Defining the Fruits of the Tree of the Knowledge of Function and Dysfunction

Now the works of the flesh are evident, which are: adultery, fornication, uncleanness, lewdness, 20 idolatry, sorcery, hatred, contentions, jealousies, outbursts of wrath, selfish ambitions, dissensions, heresies, 21 envy, murders, drunkenness, revelries, and the like; of which I tell you beforehand, just as I also told you in time past, that those who practice such things will not inherit the kingdom of God. (Galatians 5:19-21 NKJV)

We are called to bear the good fruits

Therefore bear fruits worthy of repentance. (Matthew 3:8 NKJV)

Those without faith continue to bear fruits of Discerning Function and Dysfunction. All that do this will die. Only when one repents(stops discerning function and dysfunction and has faith in God) can be healed.

so that ‘Seeing they may see and not perceive, And hearing they may hear and not understand; Lest they should turn, And their sins be forgiven them.’ (Mark 4:12 NKJV)

But he who doubts is condemned if he eats, because he does not eat from faith; for whatever is not from faith is sin. (Romans 14:23 NKJV)[Note: This is a bit out of context yet the principle concept still applies, Paul is actually talking about meat at this point of Romans.]

Example of one that will be forgiven and healed
Upon receiving the knowledge that the car has broken down, so the function {going to grandmas} is threatened. Believes that God will find a way to continue to get em to grandmas. So the display of their tree remain peaceful, loving, or joyful.

Example of one that will not be forgiven and healed
Upon receiving the knowledge that the car has broken down, judges the situation as dysfunctional. Immediately display some form of anger, depression, anxiety, blaming others for the situation, cursing creation, etc. During this change of spirit these receive their punishment and call to death until they repent.

For all can be given the Holy Spirit through Faith yet are called to remain

Abide in Me, and I in you. As the branch cannot bear fruit of itself, unless it abides in the vine, neither can you, unless you abide in Me. 5 “I am the vine, you are the branches. He who abides in Me, and I in him, bears much fruit; for without Me you can do nothing. (John 15:4-5 NKJV)

All those that do not remain in faith are taken away to receive punishment

Every branch in Me that does not bear fruit He takes away; and every branch that bears fruit He prunes, that it may bear more fruit. (John 15:2 NKJV)

If anyone does not abide in Me, he is cast out as a branch and is withered; and they gather them and throw them into the fire, and they are burned. (John 15:6 NKJV)

Conclusion
One won't be healed if they lack sufficient faith. Instead they are thrown into the fire and burned.

1 What causes fights and quarrels among you? Don’t they come from your desires that battle within you? 2 You desire but do not have, so you kill. You covet but you cannot get what you want, so you quarrel and fight. You do not have because you do not ask God. 3 When you ask, you do not receive, because you ask with wrong motives, that you may spend what you get on your pleasures. (James 4:1-3)

This above passage applies if the indicator of lack of faith is wrong (e.g. selfish) motives.

It is wrong to assume that God's decision to not heal a person is necessarily because of lack of faith. It may be because the suffering from that illness is intended by God to take a faithful person and increase their faith, as was the case with Job.

But he knows the way that I take; when he has tested me, I will come forth as gold. (Job 23:10)

Only after God's purposes in increasing Job's faith were complete did his healing come. So you could say that it was because of insufficient faith on Job's part, but not in the way that most people think. Job had more faith before his trials than many people today have who have received healing. What Job did not have was sufficient faith for the sovereign plans that God had for him, necessitating long enough and deep enough trials to grow that faith inside of him.

In the book of Acts Peter healed a lame person.

Acts 14:9-10 (KJV) The same heard Paul speak: who stedfastly beholding him, and perceiving that he had faith to be healed, Said with a loud voice, Stand upright on thy feet. And he leaped and walked.

This lame person with that faith jumped up and start walking so (I think) to be healed, you need to have faith. It all depend on the person choice. God is not even going to save one if he/she does not want to be saved.

Great question. I found a few scriptures that partially satisfy your parameters, but the only one that really suggests that miracles/healing were adversely affected by a lack of faith is below:

And when the sabbath day was come, he began to teach in the synagogue: and many hearing him were astonished, saying, From whence hath this man these things? and what wisdom is this which is given unto him, that even such mighty works are wrought by his hands?

Is not this the carpenter, the son of Mary, the brother of James, and Joses, and of Juda, and Simon? and are not his sisters here with us? And they were offended at him.

But Jesus said unto them, A prophet is not without honour, but in his own country, and among his own kin, and in his own house.

And he could there do no mighty work, save that he laid his hands upon a few sick folk, and healed them.

And he marvelled because of their unbelief. And he went round about the villages, teaching. (Mark 6:2–6 KJV)