Hi,
Well this is new !! let me try to shed some light on this !!
Before I begin let me just try to understand the question properly,This is in no way related to sound reproduction of the driver whose material is under scrutiny here before or after the damage !! if this is what you are implying with your questions then here's the answer !! if not please do rephrase the question !!
So here we go !
The paper cones are sometimes very rigid compared to the metal cones for various reasons !! one the very basic tech part,"THE MASS OF THE CONE" different companies believes in different theories !! some believe that the mass has to be lesser for appropriate sound reproduction while other believe that it has to be bit heaver ! so in general the paper cone has multilayer's when compared to the metal cones ! hence are bit more strong when it comes to physical construction ! having said that I also will say that some metal cones are thicker and stronger then paper even though they are just a single layer ( Confusing ??? well it gets better) !! in this particular case the metal cones are the winners of abuse test (if there is one ) !!
Also in general since the metal is bit stronger in terms of tensile strength etc even in comparison to multiple layers of paper or similar material,it is hence the much stronger when it comes to taking abuse ! though here the word abuse can have different meaning !!
Hope this enough !! as if I go deeper you'll get bored

!!
As you said if one were to poke the cones with a sharp object the metal cones may deform rather then allow the abject to go through creating a small hole in the cone !! where as an average paper cone easily gets punctured making a small hole !! now how this affects sound is another matter which is out of scope of this topic hence will not be dealt with !!
Now coming to the Tweeter part ! here the winner is absolutely the soft dome or non metal when compared to metal ! as most of the metal domes are very very thin,Very very delicate and sensitive( forget poking just a very small presured touch can sometimes damage these metal domes) and its very easy to damage them !! thats why if you notice most of the manufacturers who use metal domes have protective cover of some sort making it bit difficult to directly access the dome ! so as far as the tweeter goes the non metal is best !!
Hope this answers your questions !! if not please feel free to elaborate the part which is still not clear to you so that I can try to attend it !
Regards.