We should always correct over contrasts, but this is often ignored in the literature, perhaps because of the difficulty in doing so properly when there is a possibility that they are not independent.

In your case, however, as long as there is no overlap between subjects in the groups A, B and C, these 3 contrasts are orthogonal, so you can do a Bonferroni to correct over these 3, no doubt. The 4th contrast (F-test), however, isn't quite guaranteed to be independent from these other 3 and using Bonferroni for throughout the 4 may be a bit conservative, but I suspect that it is the best we can offer at the moment. This means you'd correct over 4x4=16.