Context. Recent observations of various kinds of jets in the solar atmosphere motivate studying the influence of mass flow on the stability of solar magnetic structures. Aims: We study the influence of axial mass flows on the stability of twisted magnetic flux tubes. Methods: We use the incompressible magnetohydrodynamic equations to get the dispersion relation governing the behaviour of normal modes in uniformly twisted magnetic tubes with sub-Alfvénic flows. The dispersion relation is then solved analytically and numerically to find stability criteria for twisted tubes with flow. Results: Two main important results are found. First, the axial mass flow reduces the threshold of kink instability in twisted magnetic tubes. Second, the twist of magnetic tubes leads to the Kelvin-Helmholtz instability of sub-Alfvénic flows for the harmonics with a large enough azimuthal wave number -m. Conclusions: The observed mass flow may trigger the kink instability in magnetic configurations that are near their stability threshold, leading to solar flares and coronal mass ejections. The effect is more significant for photospheric magnetic tubes than for coronal ones. Sub-Alfvénic flows undergo the Kelvin-Helmholtz instability in slightly twisted magnetic tubes if the azimuthal wavenumber is big enough.