Why can anthracene, but not phenanthrene, take part in Diels–Alder reactions? Both anthracene and phenanthrene seem to be structurally similar, with three aromatic rings. The middle rings in particular seem to be similarly activated towards cycloadditions.
Have a look at the reaction mechanism of the Diels-Alder Reaction, e.g. at Wikipedia.
We have talked about this reaction before on this site, and concluded that a sufficient explanation is only possible with the help of molecular orbitals: How accurate is this polar mechanism for the Diels-Alder reaction?
Borrowing from Wikipedia,[1] here is the most important part of the mechanism, the HOMO of the diene and the dienophile:



Now the most important part of this picture is the symmetry of the HOMO/LUMO of the diene, since this is the subject of your question. We need either a HOMO or a LUMO of these symmetries and then could match a suitable dienophile. 

In anthracene we find the appropriate symmetry of the HOMO:

The reason for the regioselectivity has also been discussed before on this site: Diels Alder with Anthracene and Maleic anhydride.
We could also tune our reaction to be of inverse demand, as the LUMO has also appropriate symmetry:


In phenanthracene we do not have the appropriate orbital symmetry in the HOMO:

The LUMO is also not suitable for an inverse Diels-Alder reaction.


Reference and Notes


"FMO of Diels-Alder reaction" by Chrito23 - Own work. Licensed under CC BY-SA 3.0 via Commons.
Calculations on the DF-BP86/def2-SVP level of theory employing Gaussian09 rev. D. Contour values for the orbitals are at 0.05&nbsp;e/rBohr. Occupied orbitals are depicted in blue and orange and unoccupied orbitals are depicted in yellow and red. Phases are chosen at random.