Organic molecules with fast non-radiative decay rates
Anything that has potential energy surfaces that cross or get close in energy. Offhand things like s-indacene and the indenofluorene family of compound, there are many others as well (azulene mentioned below). In general, if the $S_0$ and $S_1$ state for a $\pi$-conjugated organic come close in energy there is the possibility for all of that energy to be used on vibrations instead of luminescence. As far as predicive rules, that is a difficult thing to generalize. Perhaps you could argue that a low $S_0 \ce{-&gt;} S_1$ transition energy opens the possibility for non-radiative decay since these potential energy surfaces are not seperated by much energy, biradicaloid $\pi$ systems fall in this category.This is the same reason there is typically not emission from higher energy state, $S_n \ce{-&gt;} S_1$ transitions are usually quite close in energy and have crossings to get to $S_1$ efficiently through molecular vibrations. For further examples and explanations see, for example:
http://www.sciencedirect.com/science/article/pii/S0009261414008847?via%3Dihub
Or Josef Michl's excellent book "Excited States and Photochemistry of Organic Molecules" for more details on the topic.