Heavy quarkonia are observed to be suppressed in relativistic heavy-ion collisions relative to their production in p + p collisions scaled by the number of binary collisions. In order to determine if this suppression is related to color screening of these states in the produced medium, one needs to account for other nuclear modifications including those in cold nuclear matter. In this paper, we present new measurements from the PHENIX 2007 data set of J/psi yields at forward rapidity (1.2 < vertical bar y vertical bar < 2.2) in Au + Au collisions at root s(NN) = 200 GeV. The data confirm the earlier finding that the suppression of J/. at forward rapidity is stronger than at midrapidity, while also extending the measurement to finer bins in collision centrality and higher transverse momentum (p(T)). We compare the experimental data to the most recent theoretical calculations that incorporate a variety of physics mechanisms including gluon saturation, gluon shadowing, initial-state parton energy loss, cold nuclear matter breakup, color screening, and charm recombination. We find J/psi suppression beyond cold-nuclear-matter effects. However, the current level of disagreement between models and d + Au data precludes using these models to quantify the hot-nuclear-matter suppression.