jmc

As an equation, $\frac{1}{m}+\frac{1}{n}=\frac{1}{4}$. Multiplying both sides by $4mn$ to clear out the denominators gives $4n+4m=mn$. Re-arranging and applying Simon's Favorite Factoring Trick, it follows that $$mn-4m-4n+16=(m-4)(n-4)=16.$$Thus, $m-4$ and $n-4$ are pairs of factors of $16$; to satisfy the positive condition, both factors must also be positive. Then, $$(m-4,n-4)=(1,16),(2,8),(4,4),(8,2),(16,1),$$yielding $\boxed{5}$ distinct ordered pairs.