Not understanding basic C problem

gcc -Wall mytest.c
mytest.c: In function ‘main’:
mytest.c:6:7: warning: operation on ‘z’ may be undefined [-Wsequence-point]
     z = 0 || z++;

That is you hit a dark corner of the C programming language. See, for instance Sequence point - Wikipedia[^].
As a side note, enabling all the warnings (-Wall with gcc may prove itself useful, after all).

You have a logical Short-circuit evaluation - Wikipedia[^] operation of the form

a || b

The boolean result of the first expression

z = 0

is the value of the left operand z after the assignment. When that is zero, the boolean result is false. You can check this:

int z = 0;
int a = (z = 0) ? 1 : 0;
int b = (z = 1) ? 1 : 0;
printf("%d %d\n", a, b);

This should print 0 1 with all compilers.

Once the boolean result of the left term has been evaluated, the second operation should be only executed when the first was true:

int b = (z = 0) ? 1 : 0;
if (b)
    b = (z++) ? 1: 0;

Then z will be still zero because the increment operation is not executed.
[EDIT2]
The above is wrong (thank you NV)!
It should be off course if (!b)
[/EDIT2]

So compilers executing always the second operation may be not conforming to the C standard. I wrote "maybe" because I'm not really sure in this special case where the logical result is not used.

You might check it this way (all optimisations disabled; e.g. -O0 with GCC):

C++

if (z = 0 || z++)
    printf("true: %d\n", z); // Never reached
else
    printf("false: %d\n", z); // Always executed

A compiler that prints "1" here is not conforming to the C standard.
[EDIT]
Note also that optimising compilers might detect that the result of the logical operation is always false and assign zero to z or even skip the line

z = 0 || z++;

because z has been already set to zero in the previous line.
[/EDIT]

[EDIT2]
Solution 3 is pointing out the problem.
The behaviour is undefined as indicated by the warning.
Try this:

int z = 0;
z = 0 || z++;
printf ("z: %d\n", z);

(z = 0) || (z++);
printf ("z: %d\n", z);

With GCC, it will print

z: 0
z: 1

[/EDIT2]

What did you expect from the statements:

int z = 0;
z = 0 || z++;

This is again a question of whether the post-increment is performed before or after the assignment to z.