$A$ man rowing a boat upstream is at rest with respect to the shore. Is he doing work?

(NO) Work is defined as the product of force and displacement in the direction of the force $(W = F \cdot d \cdot \cos \theta)$.
In this scenario,the man is applying force to the oars to move the boat relative to the water,so he is doing work against the stream.
However,with respect to the shore,the boat's displacement $(d)$ is $0$ because the man is at rest relative to the shore.
Since $W = F \cdot 0 = 0$,the man does no work with respect to the shore.