Gluten-free formulations (GF) were utilized as food inks enriched with sour-cherry powder (SCP) and lyophilized extract (SCLE), and their physicochemical, rheological, and thermomechanical properties were evaluated with respect to different leavening conditions. Post-printing drying was also assessed in terms of texture, color, and phenolic substances. SCP and SCLE enrichment decreased lightness by 15% and increased red hue by almost 30%, mainly in yeast formulations. SC addition reduced pH by more than 12% for SCP in both leavening conditions and at 10% to 12% for SCLE, depending on leavening agents. The SCLE addition doubled dynamic moduli and complex viscosity magnitudes and increased hardness at 75.7% compared to the control for baking-powder formulations. SC enrichment, compared to the control, increased the phenolic sum to more than 90% (87% SCLE, 96% SCP) in yeast formulations, presenting lower values (almost 70%) for baking powder. Antioxidant activity in 3D-printed snacks was positively influenced by SC incorporation, depending on the drying and leavening treatment. Phenolic content, in terms of total phenolic sum, origin, and composition, possibly impacts the product’s antioxidant activity by depicting antagonistic or synergistic phenomena. Ultimately, 3D printing is feasible for producing functional GF snacks enriched with sour-cherry extracts.