There's not much known about theoretical modelling of reactions in ILs. However, as a rough starting point, you can quickly run a continuum calculation with the SMD model in Gaussian. See: JPCB 2012, 116, 9122. Going beyond that, there are force fields available for treating ILs but I'm not familiar with them. Nonetheless, a recent review like: ChemCatChem 2012, 13, 1625 may be a good start. Perhaps you can somehow implement these in a QM/MM manner for you DA reactions.
First of all you should anticipate if the IL will coordinate/interact preferentially with the diene or with the dienophile. Let’s suppose you are doing a “normal electron demand” DA, so it would be better if the IL could make the dienophile more electrophilic or enhance the electron rich character of the diene. If you are trying an “inverse electron demand” DA its just the opposite. For example if you are trying a cyclo-addition between cylopentadiene and methyl vinyl ketone, an IL which will coordinate with C=O will be favourable… If you attempting a cycloaddition for example between an electron deficient diene (or usually an heterodiene electron poor and furan other electron rich dienophiles, for example ethyl vinyl ether, an IL which may coordinate preferentially with the oxygen of the dienophiles will be prejudicial; by the other hand if it will coordinate preferentially with the diene making it more electron deficient it will be beneficial.
Summarizing, if the IL can lower the energy gap between HO MO and LUMO of the reactants it may help, if not it will be bad. Always a proper balance between stability and reactivity must be attained.
Hope I helped in some way. Good luck and good work.