Shutter time is only one of many variables that affect when dark frame subtraction would be beneficial. There is no single answer to your question.
Since what we call 'noise' is present in all digital images, it doesn't just begin to appear at some point in time. We tend to notice it when the signal (that is, the light entering the camera) is sufficiently weak enough that it is difficult to tell the effects of the light apart from the effects of the noise. Most of the time it isn't so much that the noise floor has increased as it is that the signal has decreased.
So the first variable is how strong is the signal (light) entering the camera? Of course if the amount of light is very high, we are limited to very short shutter times before the sensor begins to reach full saturation which results in a solid white image.
The next major variable is the amount of noise as compared to the signal. Things that increase the amount of noise introduced into an image:
- Heat. The warmer a semiconductor chip, such as a digital camera imaging sensor, is the more electrical signal will be generated by the effects of that heat. These electrical signals will be indistinguishable from electrical signals created by light striking the sensor. If the signal generated by the actual light is much stronger than the signal generated by heat, we don't notice the noise. But if not many photons are striking the sensor and generating a signal, then the noise will be more noticeable. We call this type of noise "read" noise or "pattern" noise.
- The random nature of light. Light, like all electromagnetic energy, travels as wave energy. Photons don't travel in a straight line. They oscillate along a wavy path. The length of the wave determines the frequency of the oscillation. This also determines the "width" of the wave. All of this vibration along the path a photon moves means the distribution of light coming from a point source is not uniformly "dense". It may be very close to uniform, and for most practical applications up until a century or so ago we could treat it as such, but at the scale of the size of most pixels on digital camera sensors, this randomness can be measured! Again, as long as there is enough light all of the randomness combines to cancel itself out and we don't perceive the effects of it. But when we are collecting very few photons with our camera sensor, the randomness of light manifests itself as what we call shot noise or Poisson distribution noise.
Dark frame subtraction does nothing for random noise. This is because the noise is different in each frame. What dark frame subtraction does is subtracts the fixed pattern noise generated in the dark frame (so-called "dark" because the shutter remains closed and the sensor remains "in the dark" as the energy generated by the sensor is measured) from the frame that was exposed to light. If automatic dark frame reduction is enabled in camera, the computation is made before the raw data is further processed and recorded to the camera's memory card for all Canon DSLRs. As far as I am aware, this is also true for every other type of DSLR that offers a similar feature.
At what exposure times do fixed pattern noise become apparent?
With enough light one can take a very long exposure and have a result with no detectable pattern noise. A properly exposed 15 second exposure using ND filters at midday to capture a waterfall will be bright enough that the pattern noise generated by the camera won't be noticeable.
On the other hand, pattern noise will become very apparent at even fairly short exposures if there is no signal to mask it. Even an exposure of, say, 1/30 at ISO 6400 with a lens cap on the camera will likely have easily detectable pattern noise. That's why I find comparative noise tests of different settings or even different cameras with the lens cap on ridiculous. The measurable noise level is only one half of the Signal to Noise Ratio. Without comparing it to a signal, just measuring pattern noise is meaningless.