On page 13 of Lack and Street's Combinatorial Categorical Equivalences, it is written (but not proven) that:
A right adjoint is conservative if and only if the components of the counit are strong epimorphisms.
I have not read their paper, but stumbled upon it as I searched for necessary and sufficient condition for a right adjoint to be conservative. I have been unable to find a reference or proof of the stated result (though another paper of Street likewise mentions it).
I would like a proof of as general a statement as possible (e.g. something like in a strict $2$-category, given a right Kan lift with counit $\epsilon$ of a $1$-morphism $Y\xrightarrow{J} Z$ that is respected by $X\xrightarrow{F} Y$ and conservative on $2$-morphisms with domain $X\xrightarrow{F} Y$, then the right Kan lift is likewise conservative on all $2$-morphisms with that domain if and only if the $2$-moprhism $\epsilon F$ is a strong epi-$2$-morphism).
I will of course also accept proof for the simple case of right adjoints being conservative if and only if the counit is a strong epimorphism, however.
I have no idea what kind of generalizations are you looking for, but the above statement is completely false even in case of ordinary categories (and you can easily find a counterexample).
It is true, however, under some additional assumptions (like, finite completeness).
Check the following question:
https://mathoverflow.net/questions/143070/two-pullback-diagram
and its answers. You may extract the proof from Lemma 2 and Lemma 3 from my note:
http://arxiv.org/abs/1311.2974
The correct versions of your statement are a kind of categorical folklore. I think that the statement was originally studied and proved (under various assumptions, including finite completeness) by John Isbell, though I do not recall any exact reference at the moment.