I understand that the variance of the sum of two independent normally distributed random variables is the sum of the variances, but how does this change when the two random variables are correlated?  
For any two random variables: 
$$\text{Var}(X+Y) =\text{Var}(X)+\text{Var}(Y)+2\text{Cov}(X,Y).$$
If the variables are uncorrelated (that is, $\text{Cov}(X,Y)=0$), then

$$\tag{1}\text{Var}(X+Y) =\text{Var}(X)+\text{Var}(Y).$$
In particular, if $X$ and $Y$ are independent, then equation $(1)$ holds.

In general
$$
\text{Var}\Bigl(\,\sum_{i=1}^n X_i\,\Bigr)=  \sum_{i=1}^n\text{Var}( X_i)+
 2\sum_{i< j} \text{Cov}(X_i,X_j).
$$
If for each $i\ne j$, $X_i$ and $X_j$ are uncorrelated, in particular if the $X_i$ are pairwise independent (that is, $X_i$ and $X_j$ are independent whenever $i\ne j$), then
$$
\text{Var}\Bigl(\,\sum_{i=1}^n X_i\,\Bigr)=  \sum_{i=1}^n\text{Var}( X_i) .
$$
