I'm Having trouble understanding which causes from an aristotelian point of view would no longer apply to them mechanistic philosophy of philosophers like Boyle, Spinoza, Galileo
For all three of them material and formal cause are important, and this was true for Aristotles predecessors.
Aristotle argued that final cause was another distinct cause, though it often collapses to the others; his main defence for it appears to be predicated on the order found in nature.
I'm not sure about Boyle or Galileo; but according to the SEP Spinoza following Descarte rejected final cause as the cause of order found in nature, and claimed it was an efficient cause - but wasn't entirely successful in removing all aspects of finalism in his cosmology:
Spinozas physical theory appears to exploit an irreducible element of finalism, and to accord an important role to individual bodily essences.
In this he differed from his main influence: Descarte who held that:
Descartes held that final cause is useless in Physics, not because physical nature is not in fact teleological but because our finite understanding cannot hope to understand the divine will, hence cannot grasp the purposes with which nature is imbued.
Just a crude answer, which you may already know.
"Final cause" is the most suspect in modern mechanical science: the idea of any teleology or ultimate purpose became very discredited in post-Baconian "natural philosophy" and the anti-scholastic Enlightenment, mocked in Moliere's "dormitive properties."
Formal and material causality are not "discredited" in the same way, but should ultimately be reducible to "efficient cause," which becomes nearly synonymous with "cause" in the mechanistic sense.
As usual, however, things are not that simple. Gravity, for example, had an uncertain "causal" status for the Cartesians, since it could not be observed to act upon objects, and Hume disputed "causality" itself on the same basis, as unobservable.
But the answer is basically: purposeful or "final" cause. Though it is now returning to scientific respectability through various approaches.
@Zedd- Just one 'side' point concerning 'materialism' and Spinoza. I was not able to quickly locate the reference in the Ethics, but I am confident that Spinoza adamantly disavowed any materialistic or mechanistic explanation of what gets termed 'matter'. For him there are two attributes; 'thought' or mind and 'extension' or body. Extension encapsulates the complete material 'world' and just as its correlative 'attribute' incorporates an aspect of 'agency-in-act'. It is not an inert substance, not 'substance at all. For Spinoza no 'final' but 'immanent' causality and it's intrinsic.