Venn diagrams are not a formal proof, nor a substitute for it, just an illustrative tool that can be useful as a guiding tool for your narrative/proof.
If writing a formal proof for this law, you will need to show
$$A \cup (B \cap C) \subseteq (A \cup B) \cap (A \cup C) \;\;\; \text{and} \;\;\; (A \cup B) \cap (A \cup C) \subseteq A \cup (B \cap C)$$
and then use the fact that if $X \subseteq Y$ and $Y \subseteq X$, then $X = Y$.
If you don't need formality, then in the appropriate context it can be used, I suppose. And, up to your ability to produce said diagrams, you could use a Venn diagram of $n$ circles, depending on what you're proving, but it gets messy quick so I wouldn't recommend it for more than $3$ sets.
In short, it depends on the level of formality that is expected of you. There's no denying that Venn diagrams in contexts like these are super, super helpful in illustrating concepts, and can be taken as a sort of heuristic proof, but they are not a substitute for formal proofs.
I say this in light of the assumption that you are probably encountering this in a class of some sort like a number of questions here. Classes in set theory, generally, will expect formality, not Venn diagrams, for example. In research, publications, journals, etc., things are much, much murkier depending on the context.
Diagram the first premise: $A \subset B$. This means that there cannot be anything inside of $A$ that is outside of $B$. In a Venn diagram, you do this by shading the area inside of $A$ and outside of $B$ ... the shading means that that area is empty:
OK, now we add to this the second premise, which is that $B \cap C = \emptyset$. So this time, the intersection of $B$ and $C$ needs to be empty, i.e. shade that very intersection. We add this to the diagram:
This diagram represents the truth of the premises. The question is now: does this force the conclusion to be true? Well, the conclusion states that $A \cap C = \emptyset$, and if you look at the diagram, we find that indeed the intersection of $A$ and $C$ is shaded, i.e. is empty. So yes, the conclusion has to be true given the diagram, i.e. given the truth of the premises. So, this is a valid argument.
For the second problem, again start with the first premise: $C \subset A \cup B$. This means that there cannot be anything in $C$ that is outside of both $A$ and $B$, and so we shade that area:
Now for premise 2: $A \cap B \cap C = \emptyset$. So, we shade the intersection of $A$, $B$, and $C$:
Now, we ask the question: does this diagram, representing the truth of the premises, force the conclusion to be true? The conclusion says that $A \cap C = \emptyset$. So, is the intersection of $A$ and $C$ empty? Well, it could be ... but there can also be something $X$ that is in the intersection of $A$ and $C$ but outside $B$:
As such, we can quickly generate a counterexample: we need to have something that is shared by $A$ and $C$, but not $B$. ... while it should still be true that there is nothing in $C$ outside of $A$ and $B$, and nothing in the intersection of all three. OK, easy:
$A = C = \{ bananas \}$
$B = \emptyset $
Best Answer
Suggestion for simplification:
An implication $p \rightarrow q$ is analogous to the set containment $P \subseteq Q$. It tells us nothing about any arbitrary element $x$, except that if $x \in P$, then $x\in Q$. Similarly, $q \rightarrow r$ is analogous to the set containment $Q \subseteq R$. So if $x \in Q$, then $x\in R$.
So you have three concentric circles, $p$ inside $q$ inside $r$ (nested), depicting $P \subseteq Q \subseteq R$. They may very well be the same circle, but none of p can be outside q, and none of q can be outside r. Nothing need be shaded. We are only given enough information to draw how the circles (sets) of the Venn Diagram are related.
That's it.