Does hierarchical agglomerative clustering with centroid-linkage suffer from chain-effect?
The potential worst cases of centroid linkage are probably too crazy to explain as easy as the single-link effect... To see how it reacts to the classic Single-Link problem, why don't you just try yourself?

Roughly, because of the way centroid linkage works, it may end up using a virtual cluster center that is outside of the actual cluster. You then may see some very weird links to happen. That is also why it can have non-monotone linkage levels (so later merges may be cheaper than earlier merges).