It depends: On the sensor design, the ISO setting selected, the length of exposure, the intensity of the light entering the camera, etc. Thermal noise can be anywhere from almost none of the noise in a digital photograph to almost all of the noise in a digital photograph.
If the photo is taken relatively quickly with moderate light intensity at high ISO settings most of the noise will be Poisson distribution noise (shot noise) that is caused by the random nature of the distribution of photons as they strike the sensor. Shot noise is not related at all to thermal considerations.
On the other hand, if the photo is taken with long exposure times and low light intensities at low ISO settings, most of the noise in the resulting image will be read noise. That is, it will be noise caused by the camera's electronics. All such noise is influenced by heat. The warmer the sensor, analog amplifiers, and digital processing units are, the more dark current they will generate that will be recorded as noise.
If a photo is taken with the lens cap on and the viewfinder covered all of the noise in the photograph will be read noise, which is affected by thermal conditions. This is just one among many reasons why testing a camera with a lens cap on is pretty much useless for predicting actual noise in real world scenarios in any meaningful way. To measure a signal-to-noise ratio, one must include signal (light) as well as noise (thermally induced dark current).
As a camera is cooled, the improvement would be more noticeable in low ISO images than in high ISO images. This is because low ISO images tend to have more dark current (read) noise while high ISO images tend to have more Poisson distribution (shot) noise. The improvement would also be more noticeable with images made using very weak light sources for longer exposure times, such as astrophotography, than with images made using very strong light sources for shorter exposure times.
Since shot noise is not affected at all by temperature, but by the very nature of light and the way photons oscillate in waves as they move, you'll never be able to have a zero noise image. In theory one could probably cool an image sensor and related electronics to the point that dark current noise would be undetectable.