For C17*, C19*, and C23*, I found that Nichon UFG2A010MDM1TD "fine gold" capacitors to be slightly better for noise than Elna Audio RFS-50V010ME3#5 "SILMIC II" capacitors. The difference in RMS noise between the two was only 3mV, which is within my measurement error. I doubt anyone could hear the difference between these two varieties. I wish I could say how these capacitors perform against more common electrolytic capacitors, but I forgot to take that data. Sorry!
For OpAmps in the output stage, I tested the RC4558, the NE5532, and the TL072. Here is the data:
RC4558 - Average Master Output Noise [RMS] = 43mV
NE5532 - Average Master Output Noise [RMS] = 40mV
TL072 - Average Master Output Noise [RMS] = 37mV
We can see that the differences in output noise are fairly small between for three OpAmps, but that the TL072 seems to perform the best. My measurement error was about 5mV RMS for these tests, so I performed a lot of averaging and took different samples at different measurement conditions in order to get as close to an ideal result as possible. What's interesting is that the OpAmp with the worst input noise spec performed the best in this test. The TL072 also has the worst input offset voltage spec! Where the TL072 performs better than the RC4558 and the NE5532 is in input bias and input offset current (and it performs much better there than the others!). This makes sense, as there was no attempt made to "cancel" the effect of opamp input current in the master section of the TR-808:

Note that the "+" inputs of the OpAmps are tied directly to ground. This means that any current to or from the "-" input will create an unequal voltage on that OpAmp terminal by itself. Of course, unequal voltage on an OpAmp's input terminals means there will be a related voltage on the output. We can see this effect in the test results with different OpAmps. It seems that less input bias current for the TL072 lead to less Master Channel noise.
Can this be improved upon? Yes, but you must be prepared to spend $5 or more on an OpAmp for this. Rather than go with the OpAmps I suggested in my other post, these results suggest that the LT1169 would be the way to go here.