Do aromatic dienes undergo the Diels-Alder reaction?
Yes, you are correct.  Aromatic compounds are stabilized by their resonance energy. This means that the activation energy for reaction of an aromatic compound will typically be greater than that for a similar reaction of a non-aromatic compound.

That said, aromatics will react as the diene component in a Diels-Alder reaction if 


the dienophile is extremely reactive (o-benzyne for example)





if the aromatic system is destabilized (for example by steric interactions or structural constraints) so that the resonance stabilization is decreased





or if the aromatic system is large enough that loss of one aromatic ring does not significantly destabilize the entire system.  Anthracene is such an example where the middle ring will react via a Diels-Alder reaction.




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