It depends on whether you want to control for the overall type I error. My guess would be yes. Multiple testing arises when you want to make sure in all 7 tests, the probability of at least one false negative is less than 5%.
However in my opinion, multiple testing adjustment is less useful in practice as it has low power. When you conduct a lot of comparisons (7 for example), you would end up with insignificant results no matter which method is used for pvalue adjustment.
To make things even worse, multiple testing happens not just in ANOVA but also in regression. Say, you fit a linear regression with 6 covariates. In the end, you will get 7 pvalues (extra one for intercept).The pvalue is calculated by comparing the estimated coefficient with zero so we actually conducted 7 comparisons. In theory, we should perform multiple testing correction. Have you seen anyone doing so? Not to mention all those model selection procedures to pick up 2 significant predictors from a pool of 50 covariates. You count how many tests are done.
It is bad that no textbook covers this. Recently there starts to have some discussion within stats society regarding this issue. No sure if there is a good solution......
My suggestion to you is to ignore this when there are too many groups. This does NOT mean you are manipulating your experiment to report a wrong pvalue that favours your conclusion. It simply means:
You cannot effectively control overall type I error in multiple comparison. However you can still control for the individual type I errors
It is valid to ignore multiple comparisons as long as you correctly interpret your pvalues.