What is the relation between $H^i_I()$ and $H^i_J()$ (cohomological functors) when $I\subset J$ are ideals of a (local) noetherian ring?

There's a map between them, and these do fit into a long exact sequence together. This is explained in the book on local cohomology by Hartshorne, see Lemma 1.8: You can even download this book if your institution has access... I will sketch it briefly. Let $V(I) = Y \supseteq Z = V(J)$ and set $W = V(I) \setminus V(J) = Y \setminus Z$, this $W$ is only locally closed in $\text{Spec} R$. We define $H^i_W(\bullet)$ to be the right derived functors of $\Gamma_W(\bullet)$. Here $\Gamma_W(M)$ for any $R$module $M$ is defined to be the set of elements of $M$ which are zero in $M_{p}$ for all $p \in (\text{Spec }R) \setminus W$. With this notation, for any $R$module $M$ we have a long exact sequence $$ ... \to H^i_Z(M) \to H^i_Y(M) \to H^i_W(M) \to H^{i+1}_Z(M) \to ... $$ Note $H^i_Z(M) = H^i_J(M)$ and $H^i_Y(M) = H^i_I(M)$. I doubt this gives you much information unless $Y$ and $Z$ have some special relationship. There are related things you can do too. If for example $Y = Z \cup X$ where $X$ is some other closed subset of $\text{Spec }R$, there's another long exact sequence but this is a special case. Can you tell me what $Y$ and $Z$ are in your case? 

