Let $X$ be a quasicompact quasiseparated scheme. Consider the full subcategory $\text{Qcoh}\_{fp}(X)$ of $\text{Qcoh}(X)$ which consists of the quasi-coherent modules which are locally of finite presentation. 


**Question** Is every quasi-coherent module $M$ the colimit of the homomorphisms $N \to M$, where $N$ runs through $\text{Qcoh}\_{fp}(X)$?


Note that this makes sense since $\text{Qcoh}\_{fp}(X)$ is essentially small. The result is well-known if we allow quasi-coherent modules which are of finite type (in particular, everything is OK if $X$ is noetherian). If $X$ is affine, then the result is trivial.



Colimit of locally finitely presented quasi-coherent modules
The answer is **yes**, at least if you believe Thomason-Trobaugh, *Higher algebraic $K$-theory of schemes and of derived categories*, which David Ben-Zvi already mentioned.


I quote from Appendix B.3 (p. 409f):



> 
> B.3. If $X$ is a quasi-compact and quasi-separated scheme, every sheaf in $\text{Qcoh}(X)$ is a direct colimit of its sub-$\mathcal{O}\_{X}$-modules of finite type. Also,
>  every sheaf in $\text{Qcoh}(X)$ is a filtering colimit of finitely presented $\mathcal{O}\_{X}$-
>  modules. ([EGA] I 6.9.9, 6.9.12.) In this case, the set of finitely
>  presented $\mathcal{O}\_{X}$-modules forms a set of generators for $\text{Qcoh}(X)$, which is then a Grothendieck abelian category and has enough injectives (cf. B.12.).
> 
> 
> 


