If a filter has a non-linear phase lag will the output signal be distorted?
I think your question comes from several misunderstandings. The fact that the phase lag of a system becomes more negative for large frequencies does not mean that there's more distortion of larger frequencies. Neither does it mean that high frequency signal components experience more delay when passing through the filter.

Imagine an ideal system that simply delays the input signal while keeping its amplitude unchanged. So for an input signal $x(t)=\sin(\omega t)$ the output signal is given by

$$y(t)=\sin(\omega(t-\tau))$$

where $\tau$ is the delay. The output signal $y(t)$ can be written as

$$y(t)=\sin(\omega t-\omega\tau)=\sin(\omega t+\phi(\omega))$$

where $\phi(\omega)=-\omega\tau$ is the phase response of the system. Note that the phase becomes more negative for larger frequencies, and this is exactly what is needed for a system that does not distort the input signal.

What causes distortion are deviations of the system's magnitude response from the value $1$, and deviations of the system's phase response from the ideal linear phase shown above.

And, as mentioned in the comments, infinite impulse response (IIR) filters like a Butterworth filter are almost always implemented in the time domain. You implement a given difference equation, which is equivalent to implementing a system with a given impulse response and a given frequency response (the latter two are Fourier transformation pairs), and the phase response is part of the system's frequency response.