Dual-supply op-amp LDO does not work, but single-supply does
First, please check LT1007's datasheet, section "Unity-Gain Buffer Application (LT1007 Only)", it has BJT input with input voltage clamp diodes. So if the output voltage goes beyond 1V + 0.7V, the input clamp diode will conduct, try to use another OP without input clamp, such as LF412, it has JFET input.

Another problem is just as others say, if your input voltage is two high, and your OP's positive output voltage swing is not high enough, then the OP can't modulate the LDO's output lower, the negative feedback loop is break then.

And the third question, if you want to make a "LDO", then the output voltage should very close to the input voltage, right? So \$V_{in}\$ should be close to \$ V_{out} = V_{ref} * \frac{R_{3} + R_{2}}{R_{3}} \$. If \$V_{in} = 10V\$, and the voltage drop on FET is low enough, the voltage feedback to the positive input of OP will be 5V or so, the OP will go to saturation, otherwise your voltage drop on FET will bigger, then it's not a 'LDO'. I suggest to lower the input voltage or change the ratio of \$R_{2}\$ and \$R_{3}\$.