The following deals only with redshift caused by motion (Doppler effect).
Wavelength (wl) shift for an object moving away from a stationary observer is calculated by following formula: shift = wl x V/C
with V speed of the moving object and C speed of light
Let's say we want a Sun-like star to become invisible to the human eye. Our Sun's emission starts around 250 nm and human vision ends around 700 nm (source: https://en.wikipedia.org/wiki/Sunlight#/media/File:Solar_spectrum_en.svg).
So we want a minimum shift of 700 - 250 = 450 nm for wl 250 nm.
Formula yields V/C = 450 / 250 = 1,8
(1) is impossible because nothing moves faster than light
(2) makes the classical formula irrelevant
The relativistic formula (for objects moving at speeds > C/10) is:
shift / wl = SQRT( (1 + V/C) / (1 - V/C) ) -1
with SQRT the square root
Rounding (shift / wl) up to 2, the relativistic formula yields V/C = 8/10
The speed V would have to be even higher for a hotter (white to bluish) star with UV emission starting below 250 nm.
So my answer to the question is:
very close to the speed of light (80% in above calculation)
and therefore not a real-life scenario (see below).
"Among the nearby stars, the largest radial velocities with respect to the Sun are +308 km/s" (source: https://en.wikipedia.org/wiki/Doppler_effect), that is 1000 times smaller than C = 3e5 km/s. Rounding down to 300 km/s, it would produce a maximum shift of 700 x 300 / 3e5 = 0,7 nm, several hundred times smaller than required for invisibility.
Besides, the star would still appear on an infrared image.