In an op-amp, what does connecting the negative supply to the feedback do?
It basically makes the LM317 a "floating" type voltage regulator. This provides the advantage that the regulator only sees the difference between the voltage input voltage and output voltage as compared to the usual style regulator that sees the voltage difference between input and ground. What this means is that the LM317 does not have a maximum input voltage, but rather a maximum Input-to-output differential voltage (see datasheet). This makes the LM317 suitable in pre-regulation application where you might have like a 48V system and you want to drop the voltage down to 20V and then use a traditional regulator to come down to 12V. This way your traditional regulator doesn't require a input voltage tolerance of more than 48V and reduces the power dissipation across it by loading it off to the LM317.
So more of less, it basically reduces the voltage seen by the opamp by basically "floating" the negative supply.