I'll try to give a "holistic" answer for what concerns the importance of noise in precision measurements, with some specific references. I'll also try to address some special topics, which are not usually covered in common texts, but which are of fundamental importance in precision measurements.
Temporary disclaimer: it will take some time to complete this answer, I hope you will forgive me. In some sense, this is a very personal answer which strictly reflects some twenty years of wandering through noise in measurements.
First and foremost, noise in all measurements constitutes a fundamental limit to the measurement resolution because it masks the variations of the measurand (the quantity subject to measurement).
We can divide this important topic in several subtopics, each of which is typically addressed by different books and papers.
Mathematical modelling of noise
Noise is modelled as a stochastic process, and this subtopic is usually addressed by books on stochastic processes. In choosing these books, it's important to take into account two aspects:
- A fundamental tool in the characterization of stochastic processes is the spectral density function: the chosen books should then have a substantial part dedicated to this tool.
- Many basic books on stochastic processes limit their treatment to wide-sense stationary processes. Unfortunately, many noise processes which are of fundamental importance in precision measurements, like flicker noise and random walk, cannot be modelled as wide-sense stationary processes. We want, then, also books that treat non stationary stochastic processes.
One of the most comprehensive, and readable, book on stochastic processes is
 A. M. Yaglom, Correlation theory of stationary and random functions, vol. I; Basic results, vol. II, Supplementary notes and references, Springer
Series in Statistics, Springer-Verlag, 1987.
For its style and wide choice of topics, this is a fantastic book (here, a review). Unfortunately, as far as I know, it's no longer available on the market. A (very) shortened version is
 A. M. Yaglom, An Introduction to the Theory of Stationary Random Functions, Dover, 1962.
The basic theory of stochastic processes is recalled also in the following book (chaps. 1-5), which is mostly devoted to spectral analysis:
 D. B. Percival, and A. T. Walden. Spectral Analysis for Physical Applications, Cambridge University Press, Cambridge, 1993.
From the same authors, the following books contains a readable introduction to non-stationary processes (§7.6-7.8):
 D. B. Percival, and A. T. Walden. Wavelet Methods for Time Series Analysis, Cambridge University Press, Cambridge, 1993.
Noise analysis of linear systems
Noise models of devices