One might experience pink noise when dealing with low frequency signals (0-100Hz). Curious to know whether pink noise has an effect on phase of the signal specially in lowest frequencies 0-10Hz.
Short answer: Yes, pink noise can and usually will effect the phase. "Pink" noise just means the power density in the noise is greater at lower frequencies than at higher. But it doesn't specify any exact frequency roll-off or signal-to-noise ratio. Also, your question depends a little on how you plan to measure the phase. If you have a pure sinusoidal signal, and measure the phase, say, by looking at a single cycle and identifying the zero crossings, then obviously any additive noise can effect the measurement. Even if you use a more sophisticated technique, like correlating thousands of cycles of your signal with complex sinusoid, any noise, pink or otherwise, can and usually will effect the phase measurement to some degree (In theory, it's possible that a particular interval of noise might exactly average out to produce no observable phase change, but that would be a fluke.) Of course, the lower the noise power and the longer the integration time, the less the effect will be.
To answer your question about "long integration times," a common way to measure phase in a communications or radar system is to "mix" the received signal with an internally-generated signal. Specifically, the received signal (for which you want to determine phase) is multiplied by a complex sine wave of the same frequency, and low-pass filtered. This produces two D.C. voltages (a "real" and an "imaginary" component), and you can determine the phase by taking the arc tangent of these two voltages. "Long integration time" means doing this multiplication/filtering over a larger number of cycles (a longer time). The longer you "integrate," the higher the signal-to-noise ratio of your result will be. That is, the noise will tend to average out the longer you go. It's a complicated subject if you haven't been exposed to it before, and I probably haven't explained it very well. There is a good mathematical description here: https://en.wikipedia.org/wiki/Phase_detector