If the residuals of a variable have a 0.6 correlation with another variable (not their residuals), I would say that this is really low, however I would always test for the significance of the corr as the ability to predict another variable through residuals is what is worrying.

If the residuals are correlated, there wouldn't be any need to report it because you will have to first correct for this before sending it anywhere for publication, imo. The correlation shows that you are missing a variable from the function that would explain the variance in the dependent variable. So you need to look for that.

Also, you have to plot the residuals individually and look whether they are centered around the fitted value (which should be flat). Any pattern here indicates that your residuals are not random, therefore there is some bias in the data.

To answer your question about what would be a threshold for correlation, I would say the closer to 0 the better. From my experience, anything below 0.3 is very low correlation and I favor this as a benchmark. So you shouldn't worry about .06.

For the benefit of, um, my friend, what is a back of the envelope acceptable threshold for correlation of the residuals? If they are only correlated .06, for example is that actually something to worry about? Can you just report the correlation?