In ANOVA, it is not an issue. In ANCOVA, it is an issue from two perspectives. In order for ANCOVA to improve the precision of an analysis, it is essential that the covariate be correlated with the dependent variable. However, if there is more than one covariate and they are correlated with each other, it is likely that you are wasting a degree of freedom diminishing the power of your design.
James E. McLean As always in multicolinearity, it depends on how strongly the two independent variables in an ANCOVA are correlated with each other. One common suggestion is that a correlation between two independent variables doesn't matter unless it is .8 or above, but I personally think that lower values than that should be considered (e.g., .6 or above). If multicollinearity is a serious concern, then it can be addressed by assessing variance inflation factor (VIF) values.
A couple of thoughts occur to me.
1) Rules of thumb about VIF cut-offs are all over the place, ranging from 2.5 to 30 in resources I have seen. The attached PDF has some slides I cobbled together on this.
2) If you believe Paul Allison, how much multicollinearity one can tolerate depends on what one is doing. If one's primary interest is in prediction, multicollinearity might be largely irrelevant. But if one is interested in precise estimation of the coefficients for individual predictors, then multicollinearity can be quite problematic. See Allison's post on this topic here:
https://statisticalhorizons.com/prediction-vs-causation-in-regression-analysis/
So, Dory Dee, which of Allison's two approaches to modeling matches more closely what you are doing?
Another thought about multicollinearity in ANCOVA: what do you mean with it? The correlation between the IV and the CV? For ANCOVA to work best, IV and CV should be uncorrelated (what should be the case with random allocation to the IV groups). Otherwise, parts of the variance between the IV groups cannot be attributed to the IV, if it is correlated with the CV. That means that you may reduce the power to detect your IV effect (which is clearly not what is intended with an ANCOVA). Is that something you mean or have considered?
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