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 O2(w) at point z gives :
O2(w)=O2(z)+(w−z)∂zO2(z)+...
taking derivative wrt w on both sides we get
∂wO2(w)=∂zO2(z)+...
Using these two results in your first equation we get
O1(z)O2(w)=O2(z)(z−w)2+regularterms
Subtracting it from your second equation, multiplying with (z−w)2 and taking limit w→z we conclude that O2 and O1 should be equal. Since to begin with we didn't assume any such thing regarding fields O2 and O1 so in general your second equation shouldn't follow from the first one.
I think equality of O2(w)O1(z) and O1(z)O2(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