Inspired by recent experiments on chromosomal dynamics, we introduce an exactly solvable model for the interaction between a flexible polymer and a set of motorlike enzymes. The enzymes can bind and unbind to specific sites of the polymer and produce a dipolar force on two neighboring monomers when bound. We study the resulting nonequilibrium dynamics of the polymer and find that the motion of the monomers has several properties that were observed experimentally for chromosomal loci: a subdiffusive mean-square displacement and the appearance of regions of correlated motion. We also determine the velocity autocorrelation of the monomers and find that the underlying stochastic process is not fractional Brownian motion. Finally, we show that the active forces swell the polymer by an amount that becomes constant for large polymers.