When a conservative force does positive work a positive work on a body, how does it affect the potential energy of the body?

increases its potential energy

hey, 

the potential energy will remain unchanged. 

the potential energy of a system is defined corresponding to INTERNAL CONSERVATIVE FORCES and is equal to negative of work done by them. The only force acting on the body (in the question) is an external force and hence potential energy change is not defined or simply potential energy will not change. only kinetic energy will change (acc to work energy theoram).