What are some of the strongest theories against the existence of mantle plumes?
The best argument I've heard supporting strong skepticism of plumes, if not total dismissal, is that the theory is too flexible. To put it more bluntly, this amounts to saying that it is unfalsifiable and therefore not helpful (in Popper's words, "A theory that explains everything, explains nothing.").

Erik Lundin, a Norwegian geologist, is a fairly vociferous (and sarcastic) proponent of this view. He likes to point out that you can explain almost any tectonic anomaly with plumes. Have a look at his articles The Iceland “Anomaly” – An Outcome of Plate Tectonics and Phanerozoic fold belts and volcanic passive margins: a causal relationship? on Gillian Foulger's site mantleplumes.org — where you'll find lots of other evidence for and against plumes. 

My own view is that there seems to be some clear cases of plume-like things being a good explanation for observations — the Hawaiian–Emperor chain comes to mind. And then there's the rest of the infinitely complex earth, with thousands of competing and interacting forces, convection currents being just one.