Relation between $H^i_I(-)$ and $H^i_J(-)$ when $I\subset J$  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?
 A: 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?
