For $A,B$ and $ C $ partially pairwise independent occurrences (i.e. $I(A;B)=0$, $I(A;C)=0$ ), it is not true to say that $I(A;B,C)=0$, since $I(A;B,C)=I(A;B)+I(A;B|C)$ [<-this is not correct, see edit] and we have no information about $I(A;B|C)$.
If i had to use venn diagram on the above case, i would use this diagram, in which $I(A;B,C)=0$.
I use venn diagram often when I need to prove/disprove equalities and inequalities in information measures, but my intuition seems to mislead me here.. So my question is - is it possible to draw an appropriate diagram for this case? or should i stop using such diagrams since they do not fit all cases?
EDIT: i have made a mistake when i wrote down my question here, possibly misleading some of the answers. the correct identity is $I(A;B,C)=I(A;B)+I(A;C|B)$ (note its A;C|B and not A;B|C). when wrote in the correct way, it is possible to use the inequality ("conditional reduces entropy") and get $I(A;B,C)=I(A;B)+I(A;C|B)\le I(A;B)+I(A;C)=0$ (the last equality comes form the question terms). this only confuses me more, since i did not expect $I(A;B,C)$ to be zero (again, pairwise independent between A,B and A,C).
EDIT2: Im not sure about this inequality $I(A;B)+I(A;C|B)\le I(A;B)+I(A;C)$. if anyone can help out, ill be glad. I will update when i will be sure what happens here.