Herein, a series of CuI or CuNCS complexes with neocuproine (2,9-dimethyl-1,10-phenanthroline: dmp) and two tris(aminomethyl)phosphines derived from morpholine (P(CH2N(CH2CH2)2O)3) or thiomorpholine (P(CH2N(CH2CH2)2S)3) were tested as cytotoxic agents in vitro towards mouse colon carcinoma (CT26) and human lung adenocarcinoma (A549). The studies showed that the complexes exhibit potential antitumor properties, displayed by IC50 values below 10 μm towards the tested cell lines, in the case of 4-h incubation time with the examined compounds. Moreover, a high antimicrobial activity of all the complexes was observed against Staphylococcus aureus and Candida albicans with minimal inhibitory concentrations equal to 1–2 μg/mL. To gain insight into the molecular mechanism of biological activity of the complexes, we investigated also their interactions with plasmid DNA (pUC18) and the human and bovine serum albumins. Gel electrophoresis experiments demonstrated that all the compounds were comparably efficient in DNA degradation process; however, luminescence quenching showed surprising dependence on the interactions strength of the used compounds with the albumins. Apart from exceptionally effective [CuI(dmp)P(CH2N(CH2CH2)2O)3], the complexes with P(CH2N(CH2CH2)2O)3 quenched more strongly luminescence of bovine serum albumin, while the complexes with P(CH2N(CH2CH2)2S)3 were more active in the quenching of human serum albumin luminescence.