How much algebraic geometry is there in complex geometry (for example, Demailly)? I wonder how much of modern algebraic geometry (schemes, etc) is there in complex (algebro-analytic) geometry. What I mean is complex algebraic (analytic) geometry in the sense of Demailly, Lasarsfeld etc.
I wonder if scheme theory (as in Grothendieck's work) is actually useful there. 
 A: You  don't need any scheme theory at all to work in complex geometry.    
I  know, because I started doing research in complex geometry and didn't know any scheme theory, nor actually any algebraic geometry at all.
In fact it might be the other way round: sheaf theory and  cohomology are much more down-to-earth in the context of manifolds and once you understand them there  you may try to transport your technical knowledge to the much less intuitive theory of schemes built up from  arbitrary rings, including incomprehensible monsters like $(\mathbb Z/2\mathbb Z)^\mathbb N$.    
Historically most of the advances in complex geometry were done by people who didn't know scheme theory, in particular because it didn't exist at the time !
And again it is the other way round: complex geometers like Teichmüller, Oka, Stein, Kodaira, Hirzebruch, Serre (in his early fifties period), H. Cartan were at the cradle of scheme theory (although the history is much more complex [pun intended]).
Of course there are domains that mix up both approaches (say positivity of vector bundles) and some brilliant practitioners master both (Demailly, Griffiths, Kollár, Lazarsfeld, Ueno,...) but  Fields medalists like Ahlfors, Kodaira, Hörmander,  Fefferman did splendid work in complex analysis without ever writing the word "scheme".
And ditto for Abel Prize laureates like Nirenberg or Singer.
So if you want to really learn complex geometry immerse yourself into elliptic operators, Hodge theory, plurisubharmonic functions, pseudoconvex domains and let the EGA on their  shelf, for the time being at least :-) 
A: I am not a complex geometer, but my understanding is that since complex geometry is a vast and old field of research, the blend of methods (analysis, differential and algebraic geometry, topology) varies considerably depending on the topic.
To complement Georges Elencwajg's answer, let me mention the following. As far as scheme theoretic ideas go, they appear in the following elementary foundational topics (judging from what I have read in the standard textbooks of Demailly, Voisin and Griffiths-Harris):

*

*Chow groups and intersection theory on smooth projective varieties. Apart from studying intersection theory in a purely topological (cup products in cohomology) and differential topological ways (wedge products of differential forms), it is also important to understand the algebro-geometric Chow groups. Voisin in Chapter 9 of Volume II mainly follows Fulton's treatment, which is purely algebraic and heavily uses scheme-theoretic machinery (multiplicities, generic points, nilpotent, etc.)


*Deformation theory. It appears that the algebraic approach to deformation theory of complex manifolds and vector bundles is also pretty commonly used in complex geometry. E.g. in Chapter 9 of Volume I, Voisin uses infinitesimal neighbourhoods and rings of dual numbers to explain the Kodaira-Spencer map.
At the research level, it appears that complex geometry is even more interconnected with algebraic geometry. It is hard to tell which way the flow of new ideas go, e.g. multiplier ideal sheaves come to mind, and recent research on the interaction of various positivity notions in its differential- and algebro- geometric incarnations (Demailly, Lazarsfeld).
A: Oka's work in complex geometry used "ideals of indefinite extent" (or similar wording) which were operationally sheaves of ideals.
And as other comments and answers have indicated, at the very least, the short-to-long exact sequences of sheaf theory give a very clean explanation of Chern classes and other basic things. R. Gunning's yellow Princeton book on Riemann surfaces gives a very persuasive introduction to this (though using a "covering" version of sheaf cohomology rather than "derived functor of global section functor", but this is relatively a secondary detail).
