This study investigates how layer deposition in printing 3D models made of polyetherimide (PEI) using the fused filament fabrication (FFF) technique affects resistance of these models to abrasive wear. Samples made by additive manufacturing with different setting of build orientation and path generation strategy of extruded fibers were used in the experiment. The experiments were conducted on a tribometer according to the ASTM G65-16 standard. The friction force, normal force, and temperature in the contact area during the tribometer operation were measured using a strain gauge sensor. The tribometer allowed us to perform the tribological experiments using a rubber-coated or a metal disc without a lubricant in the so-called “dry” operation. Following the ASTM G65-16 standard, a rubber-coated disc and Fe3Al2 (SiO4)3 garnet abrasive were used for the wear resistance tests. The analysis of experimental data showed correlations among orientation selection, the strategy of layer deposition by the FFF technique, and the surface life in terms of abrasive wear. The results of this research also showed the suitability of the chosen building orientation and deposition strategy for part production by additive manufacturing, depending on the required tribological properties, such as the coefficient of friction (yield path) and wear behavior. Based on the results of the study, it is concluded that continuity of wear and friction force depends on the path traveled under the model production orientation. The size of wear (material loss) ranged from 0.451–0.809%. It was shown that the weight loss of the sample under loading was greater, on average, with the chosen fiber orientation strategy in the Z direction than in the X direction.