Considering Long exact sequence of cohomology Let $X$ be a projective scheme over a field $k$.
If $0 \longrightarrow F' \longrightarrow F \longrightarrow F'' \longrightarrow 0$ is a short exact sequence of coherent sheaves on $X$, can I move to a long exact sequence of cohomology
$... \longrightarrow H^i(X,F')\longrightarrow H^i(X,F)\longrightarrow H^i(X,F'') \longrightarrow  H^{i+1}(X,F') \longrightarrow ...$ ?
How can I justify this? Where do I find justification in Hartshorne for this?

Yes, this is true.
It is a general fact of homological algebra, that if an abelian category $\mathcal C$ has enough injectives, then derived functors are defined, and short exact sequences in $\mathcal C$ yield long exact sequences of derived functors. See Hartshorne's Algebraic geometry, Chapter III Theorem 1.1A.
Now the sheaf cohomology groups $H^i(X, F)$ are defined to be the derived functors of the left-exact functor $\mathfrak{Ab}(X) \to \mathfrak{Ab}, F \mapsto \Gamma(X,F)$. This is the definition in Chapter III, 2 Cohomology of Sheaves.
