I often find parts of financial correlation matrices not statistically significantly different from zero
You might want to be careful with that. First of all, statistically significant and large enough to be meaningful are not the same thing. Second, be careful that the test you use to assess the significance of the correlation coefficients do not assume the underlying data to be Gaussian because if you are dealing with financial series, they are most likely fat-tailed (and this is a case where it would make a difference, i.e. render your test void). Third, testing whether a single correlation coefficient is statistically significant is not the same as testing whether some of your correlation coefficients are statistically significant.
Also, the sign of correlations can skew the results.
Presumably, not if the correlations are small to begin with (i.e. if you only fudge with small correlation coefficients this should not be a concern).
see Edward's comment below.
fix the negative eigenvalues that might arise due to such corrections?
decreasing the absolute value of a correlation coefficient (as you consider doing) will never make a PSD matrix become indefinite (<=> it will not change the sign of any of your eigenvalues).
see Whuber's comment below.