Elastic stress and strain can develop within living tissues, caused by processes like cell proliferation and apoptosis. Besides, the stress caused by tissue growth relaxes over time leading to an intricate coupling between mechanics and shape dynamics. To better understand this process, we introduce a computational phase-field method to simulate a growing tumor in a surrounding tissue taking elastic as well as viscous properties into account. We couple continuum modeling of the viscoelastic mechanics to the concentration of a diffusible growth-promoting nutrient. The implicit description by a phase-field variable is a very stable and flexible way to describe the moving interface between the tumor and the surrounding tissue. We study the effect of differential growth in two and three dimensions for various mechanical parameters and observe not only stationary shapes but also finger formation during tumor evolution.