Do eta-4 benzene ligands participate in Diels Alder or other cycloadditions?
Coordinated benzene as a dienophile

It would be logical to suggest that an η4-benzene ligand could behave as a dienophile. The reason why benzene is characteristically unreactive, both as a dienophile and as a diene, is because of its aromaticity; if aromaticity is lost, one would expect it to become more reactive.

However, to the best of my knowledge (and SciFinder-fu), such reactivity does not seem to have been achieved. :(

In the rest of this answer I'll try to summarise some of the research that is (somewhat) related to what you were asking about.



Benzene as a dienophile - in general

Benzene itself (and some substituted benzenes) can act as a dienophile under some conditions:


with 1,2,4,5-tetrazines which act as electron-poor dienes (i.e. inverse demand Diels–Alder).1 Presumably, the reaction is favoured due to the loss of $\ce{N2}$ gas.

Under high pressure.2

Intramolecularly, when constrained to be in close proximity with a reactive diene.3

With incredibly reactive dienophiles (cyclobutadienes).4



(This list is probably not exhaustive.)



Coordinated benzene, but as a diene

In ref 4, Inagaki et al. also summarised the ways in which benzene has been activated towards Diels–Alder reactions:


  Currently, there are only a limited number of options to activate benzene sufficiently, including coordination of a transition metal complex to benzene, naphthalene with small resonance energy and the use of benzene derivatives with a sterically induced strained geometry [...] (emphasis mine)


The only reference given for the bolded sentence is the same article that Klaus has linked in his answer.5 It describes the reaction of η2-benzene as a diene, with a hypothesis very similar to what you have written in the question:


  In the proper coordination environment, both osmium and rhenium can form stable complexes with arenes in which only two of the six carbons are coordinated. Once complexed in a dihapto fashion, the uncoordinated portion of the arene more closely resembles a conjugated diene. We therefore hypothesized that the complexation should increase the propensity of arenes to undergo cycloaddition reactions.


In this reaction scheme taken from the paper, the carbons numbered 1 through 4 are acting as the diene component in a Diels–Alder reaction with N-methylmaleimide:





References


Seitz, G.; Hoferichter, R.; Mohr, R. Benzol und linear anellierte Arene als Dienophile in der Diels-Alder-Reaktion mit inversem Elektronenbedarf. Angew. Chem. 1987, 99 (4), 345–346. DOI: 10.1002/ange.19870990413.
Jarre, W.; Bieniek, D.; Korte, F. Benzene as Dienophile in the Diels-Alder Reaction. Angew. Chem. Int. Ed. 1975, 14 (3), 181–182. DOI: 10.1002/anie.197501812.
Saito, K.; Omura, Y.; Maekawa, E.; Gassman, P. G. The Addition of 1,3-Diphenylisobenzofuran to Cyclooctatetraene and to Dimethyl Cyclooctatetraene-1,2-dicarboxylate. The Benzene Ring as a Dienophile in an Intramolecular Diels–Alder Reaction. Bull. Chem. Soc. Jpn. 1990, 63 (2), 395–400. DOI: 10.1246/bcsj.63.395.
Inagaki, Y.; Nakamoto, M.; Sekiguchi, A. A Diels–Alder super diene breaking benzene into C2H2 and C4H4 units. Nat. Commun. 2014, 5. DOI: 10.1038/ncomms4018.
Chordia, M. D.; Smith, P. L.; Meiere, S. H.; Sabat, M.; Harman, W. D. A Facile Diels−Alder Reaction with Benzene: Synthesis of the Bicyclo[2.2.2]octene Skeleton Promoted by Rhenium. J. Am. Chem. Soc. 2001, 123 (43), 10756–10757. DOI: 10.1021/ja011689q.