If you are asking if you can carry out all pair-wise comparisons via Tukey's HSD test when the omnibus F-test from the ANOVA model is not statistically significant, the answer is YES,you can.  Tukey's HSD has no requirement of a significant omnibus F-test.  The main reason for computing the ANOVA (in the old days especially) is that one needs the MS_error from the ANOVA model to compute the test statistic values.  Among the most common multiple comparison procedures, the only one that requires a statistically significant omnibus F-test as a first step is Fisher's Least Significant Difference (LSD) procedure.  HTH.