Assume X and Y are two positive RVs and Cov(X,Y)>0. Does this imply that Cov(X,1/Y)<0, or is more information needed?
No, this is not implied. The sign of a covariance is essentially only preserved in a consistent way by linear transformations: for all other functions, including f(x)=x−1 you can exploit the curvature of the function to make the sign whatever you want.
Here's a quick example I got by playing around with the numbers:
suppose you sample (1,1), (2.5,0.1) and (3,2) uniformly to generate (X,Y) pairs. This gives positive covariance, and still does if we replace the Y values by 1,10,0.5.
There may be numerically simpler examples available, but at least three points are necessary.