Suppose that an object $O$ moves in the plane $x,y$ along a path with respect to time $ t $ of the form $O (t) = (x (t), y (t)) = (2 \cos (t) , 2 \sin (t))$ and another $ P $ object while moving along a path $ P (t) = (z (t), w (t)) = (r \cos (t + r ), \sin (t + r)) $ with $ t $ and $ r $ real numbers.
How can I find the minimum and maximum distance between both objects as function of $r$.
I assume you mean Euclidean distance. Recall that the square of Euclidean distance between two points is given by: $$ F(t,r) = ||O-P||^2 = (x-z)^2 + (y-w)^2 = \dots $$ Now find extrema of $F$ treating $r$ as a constant parameter (simple calculus will do).
The minimal and maximal distance between objects can then be described as a function of r: $$ L_{\max}(r) = \sqrt{F(t_{\max},r)} \qquad\qquad L_{\min}(r) = \sqrt{F(t_{\min},r)} $$