At what speed does something have to travel away from us for it to red shift enough that it becomes invisible to the human eye? Are there stars, galaxies etc that we cannot see because they are traveling too fast and their spectrum is shifted below our visible range? From what I understand, red shift is caused by stars and such moving away from the viewer. At what speed does a star etc have to travel to be invisible to us?

What if you are in one galaxy on one side of the universe that is expanding looking at a galaxy on the other expanding edge side. They would be traveling faster than the speed of light away from each other. Would they be invisible to each other?
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
(source: https://en.wikipedia.org/wiki/Doppler_effect)

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
which
(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
(source: https://en.wikipedia.org/wiki/Relativistic_Doppler_effect)

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.