I don't want to make this sound like speaking latin in the church so common people can't understand.
Taken very simply, you can generally replace a JFET input opamp, with a newer, better JFET opamp and not make anything worse, but it might not make things much better. Likewise, a bipolar opamp should be easily updated with a newer bipolar opamp. The general consideration is input bias current. FET input opamps have less bias current, but modern bipolar opamps are much better at canceling out this bias current, so some advanced modern bipolars might work in a Bifet or JFET input socket (or not).
Opamps are general purpose gain blocks with huge open loop gain. The feedback network applied around an opamp trades a bunch of that open loop gain to deliver a more precise closed loop gain as defined by that feedback network. Beyond a point, adding more open loop gain, has a diminishing effect on linearity. Likewise, wrt opamp speed, if the opamp is already faster than the signal being passed, substituting a faster opamp will not make a difference, because the audio signal isn't trying to move that fast.
Even these comments are a gross simplification, and better is usually better. My suggestion about needing to redesign feedback values was for the specific case of some modern ultra low noise opamps. When the opamp noise gets that low, the thermal noise of the feedback network now becomes a limiting factor. So to get the full S/N capability of the new opamp, one needs to substitute smaller value resistors that make less self noise. But now everything draws more current, so a marginal layout for stability may no longer be stable and power supplies could run out of current.
I don't want to make a blanket dismissal of all component upgrades, and products designed before what I consider truly audio capable opamps were available, can surely benefit, but perhaps ironically the 553x and TL07x listed as targets to swap out are what I consider the newer generation of audio capable opamps, with speed well in excess of what is needed for audio and decent noise performance. The older opamps than those were truly marginal and more difficult to apply for good results.
For difficult applications (like high gain), or where the original execution was marginal.. (like using a TL07x for an output stage), there may be some room for improvement. Otherwise when properly applied the 553x and TL07x should not be the limiting factor for signal integrity.
So I'm sorry, there is no simple set of guidelines. The opamp is a general purpose part, that can be made to perform a very large universe of very different applications, so any simple rules would have to be limited to very specific applications.
IMO many such swaps are a waste of time and money, but that means not all are. So YMMV.
JR