IMO intelligence is a biologic function no different from any other, some animals have more of it than others, roughly dependant upon the brain to body mass ratio. Saying that intelligence is definitively not a product of natural causes would require some evidence to support that position if the claim is to be a scientific one, but not if it is a religious claim.
(1) Can evidence ever be produced for non-natural causes?
Actually, it is science - as generally accepted today - that asserts that by definition it can accept none other than natural causes.
That rules out even listening to any explanation with "other than natural causes". Its a total waste of time providing them.
The game is fixed in advance --- if anybody tries to advance a viewpoint that an objectively true statement about reality can contain other than 100% natural causes -- he is ruled out of order.
You may also be one who agrees that reality consists only of natural causes - and other explanations are not acceptable, but you are agreeing with a philosophical statement, not a scientifically proven fact.
(2) Is it necessarily "religious" to believe that something is objectively true and at the same time not material?
I'd say there a lot of things that a) exist,
are objectively true and c) are not part of the natural material world.
We have just been brainwashed , er conditioned, to believe that scientific inquiry is a self-standing structure with no non-natural underpinnings.
There are plenty of real immaterial things. For example: the rules of logic... the existence of a mind ... and core existential idea set ( i.e. the common perspective that a universe outside one self exists)
These collections of reality can neither be proven or confirmed by natural means because they are outside the material world.
These realities are in fact assumed by science and used by science ... they are not proven by science.... they stand outside science... and are NOT false.
Or... go ahead and asset they are all false ... and after that you have no warrant to make a single scientific statements of any relevance or merit.
(3) What would lead one to say that intelligence is not a physical phenomenon?
If ideas are just "secretions of the brain", there is no reason to find it plausible that a mindless purposeless random process could produce RELIABLE ideas simply as secreted by randomly-produced skull meat.
If true ... science, with all its protocols and rules, its findings and all that .... is just random bunkum secreted by the same skull meat.
The most cherished "facts" carry no credibility - delusion and vanity makes us believe that we have a grasp of anything.
If ideas really are a secretion of the brain ... The core delusion then becomes not the religious ones (e.g. that the soul exists), but the scientific delusion that reliable knowledge exists in science.
(4) What facts make it clear that the mind is not just a physical phenomenon?
First let me say how appealing it would be if our minds were machines ... our acts would be deterministic and we are absolved of any need to follow rules, blame for error, hurt, or immorality. In fact, we could be truly free at last because we are slaves to a rotting meat machine.
That is, it seems appealing FOR ONE SELF until we apply the same "benefit" to the rest of humanity that deals with us as they wish. This is truly diabolical.
In spite of the many obvious paradoxes this line of reasoning gets us into... ... That's only one side to consider.
Let's go back to the question again.
The mind could conceivably be entirely a physical phenomenon ... but it would be very unlike other physical phenomena.
Take one experiment... blink your eyes. You can repeat it because you mind controls the activity.
Now shut off your mind.
You can't do that... but are capable ... you dream at night and shut off awareness.
Now repeat the idea you just had exactly as it was.
You can't do that... because you mind "moves on" and can only amass additonal bits of ideas - it cannot go back to just where it was.
The mind seems to be capable of controlling other physical actions... so if it too is physical we need to explain why it is exempt from this control.
So why is it exempt from control? NOT because it is involuntary like a heartbeat that cannot be directed ... we know we can direct our minds by providing stimulus such as a book or a computer web page...
But what ever causes us to give our mind that direction? Zero! Since we are compelled deterministically by the mind to direct the mind on to a subject for stimulation,
That's contradictory. .
As soon as you asset that the mind is just a mere physical machine, you generate paradoxes and no insight. Either the assertion is faulty ... or paradoxes are all the reality you will ever be able to know.