Now the thing is that the given expression is zero if and only if $x,y,z$ are linearly independent ...ie when $r$ is a free position vector... Now if $r$ is the position vector on a curve or a surface then $x,y,z$ are not independent .. Rather they are dependent and the answer shall never be zero..
Example if $r$ is a position vector of a point on the surface $x²+y²+z²=1$.. Which is the equation of a sphere.. There is obviously curling or rotation and definitely position vector is not zero..
The curl operator applies to vector fields it doesn't make sense for individual vectors. the vector field you are interested is just a map $F:\mathbb{R}^3\rightarrow \mathbb{R}^3$ with $F=(F_1,F_2,F_3)=(F_1(x,y,z),F_2(x,y,z),F_3(x,y,z))$.
Let's break down this notation. For each coordinate $(x,y,z)$ the vector we end up getting for the $x$ coordinate, $y$ coordinate, and $z$ coordinate each individually depend on three functions. $F_1$ inputs the $x$ coordinate of the vector (and then outputs an appropriate vector), $F_2$ inputs the $y$ coordinate of the vector and $F_3$ inputs the $z$ coordinate. Combining this three, we obtain the vector $F$. Inputting every vector $(x,y,z)$, we obtain the vector field we are interested in. The specific vector field we get depends on the individual functions.
If we compute the curl of a general vector field, we end up getting $(\partial F_z/\partial y-\partial F_y/\partial z)\hat i +(\partial F_x/\partial z-\partial F_x/\partial z)\hat j -(\partial F_y/\partial x-\partial F_x/\partial y)\hat k$
The first component of your vector field is $x$, the partial derivatives vanish, as $F(x)$ is independent of $y,z$. The same happens for $F(y)$ and $F(z)$. So the curl of our vector field is just the zero vector $(0,0,0)$, and so $curl(F)=0$
More generally, if the first coordinate of your vector field only depends on $x$, the second only depends on $y$ and the third only depends on $z$, the curl will be zero.
If you want to think about something interesting relating to curl, consider the following:
If your vector field is of the form $F=(F(x), F(y), F(z))$, the curl will be zero, however there are other ways to obtain zero curl. Can you find these examples? Can you find a general rule for when the curl of a vector field would be zero?