Photodiode amplification works with LM258N but doesn't with LF356N
LF356 is not a single-supply op-amp - ie. the input common mode range (and output swing with a load to the negative rail) do not include the negative rail. The CA3140 and LM358 are single-supply op-amps. 

The op-amp in your circuit can only be in balance when both inputs are at the negative rail, hence it cannot work with the LF356. As you can see from the datasheet the op-amp needs the inputs to be at least 4 or 5V above the negative rail to be guaranteed to work properly. The output swing is a similar problem- it cannot approach the negative rail closer than a few volts, depending on the load. 



If you add a negative supply (eg. -5V), just for the op-amp negative rail, it will work properly.