Collagen networks form the structural backbone of the extracellular matrix in both healthy and cancerous tissues, exhibiting nonlinear mechanical properties that crucially regulate tissue mechanics and cell behavior. Here, we investigate how the presence of invasive breast cancer cells (MDA-MB-231) influences the polymerization kinetics and mechanics of collagen networks using bulk shear rheology and rheo-confocal microscopy. We show that embedded cancer cells delay the onset of collagen polymerization due to volume exclusion effects. During polymerization, the cells (at 4% volume fraction) cause an unexpected time-dependent softening of the network. We show that this softening effect arises from active remodeling via adhesion and contractility rather than from proteolytic degradation. At higher cell volume fractions, the dominant effect of the cells shifts to volume exclusion, causing a two-fold reduction of network stiffness. Additionally, we demonstrate that cancer cells suppress the characteristic stress-stiffening response of collagen. This effect (partially) disappears when cell adhesion and contractility are inhibited, and it is absent when the cells are replaced by passive hydrogel particles. These findings provide new insights into how active inclusions modify the mechanics of fibrous networks, contributing to a better understanding of the role of cells in the mechanics of healthy and diseased tissues like invasive tumors. Statement of significance: Understanding how cells influence tissue mechanics is crucial to unravel disease progression. While fibroblasts are known to stiffen tissues, the role of invasive cancer cells is less clear. Using collagen-based tissue models, we reveal that cancer cells unexpectedly soften the collagen matrix and disrupt its stress-stiffening response. By comparing active cells to passive particles and selectively blocking cell functions, we show that volume exclusion, adhesion, and contractility each play distinct roles in shaping tissue mechanics. This work sheds light on the physical impact of cancer cells on their environment, advancing our understanding on how cells dynamically alter the mechanical properties of tissues.