Do all 'or' cases need to be checked in a direct proof? There's a problem from Richard Hammack's book of proof, which asks to prove the proposition: 

Suppose $x,y \in \mathbb{Z}$. If $5 \nmid xy$, then $5 \nmid x$ and $5 \nmid y$

In this case, the intent is to show the use of the contrapositive proof. So a direct proof on $$5\mid x \lor 5\mid y \implies 5 \mid xy$$

The proof is straightforward, and in the book the solution shows that when $5\mid x$ is true, then $5 \mid xy$ is true, and similarly for $5\mid y$. 

In this case, the solution doesn't show the case where both $5\mid x$ and $5\mid y$ are true. 

In this particular example the proof of the third case is trivial, and so I understand that it can be left out entirely.

However, I am wondering in general, when using direct proof, and when the $P(x,y)$ statement is compound statement made up of two statements conjuncted with an or, then is it necessary to check all 3 cases? 

In other words, to link it back to the example proof, is it strictly necessary to show that $5\mid x$ AND $5\mid y$ to complete the proof? Or is it that since we have shown the T/F and F/T case, the T/T case is also always true?
The case $5|x$ and $5|y$ is already covered by the other cases, so it is not necessary to consider it separately. The important thing is to make sure that the cases you consider are exhaustive.
Consider it this way: If you have separately proven that $5|x$ is true and $5|y$ is true, you can linguistically say that you have proven that the statements $5|x$ and $5|y$ are both true. This is logically the same as $5|x \wedge 5|y$ being true, so you do NOT need to separately check this, since you have already done so logically. Typically, the "and" statement is not considered a separate case, so a proof by cases comprises only of proving the separate statements.