I am not an expert in 2d CFT. However I hope following manipulations are valid.
Assume that your second equation follows from first one.
Then on RHS of your first equation Taylor expansion of $O_2(w)$ at point $z$ gives :
$O_2(w)=O_2(z)+(w-z)\partial_z O_2(z)+ ...$
taking derivative wrt $w$ on both sides we get
Using these two results in your first equation we get
Subtracting it from your second equation, multiplying with $(z-w)^2$ and taking limit $w\rightarrow z$ we conclude that $O_2$ and $O_1$ should be equal. Since to begin with we didn't assume any such thing regarding fields $O_2$ and $O_1$ so in general your second equation shouldn't follow from the first one.
I think equality of $O_2(w)O_1(z)$ and $O_1(z)O_2(w)$ (assuming fields are 'bosonic') within time ordered product only implies that their OPE should be symmetric under exchange of z and w. So if your first equation for OPE can be realized for some (bosonic) fields, then by exchanging z with w on RHS you should get the same result within a regular term.
This post imported from StackExchange Physics at 2015-03-30 13:50 (UTC), posted by SE-user user10001