Suppose I want to see whether $z$ is a confounder for a model with $y$ the outcome variable and $x$ the predictor. If I adjust for $z$, and the adjusted coefficient of $x$ changes versus the unadjusted coefficient of $x$, does it matter by how much it changes? If the difference between the unadjusted and adjusted coefficients for $x$ is very small, should I still take this into account?
"How much" matters a great deal! The adjustment is unlikely to be zero, after all; this would only happen if z were totally uncorrelated with x or y.
By common convention one would test the statistical significance of the relationship between z and y as a way of deciding whether it is necessary to use z to adjust x's coefficient. That said, significance will depend on other things such as alpha and sample size. You may find yourself having to make a largely subjective judgment as to whether to adjust, based on some combination of p-value and the magnitude of the proposed adjustment. Which may be ok, as long as you document the factors that went into your decision.
In practice, small adjustments are likely to matter more in domains such as pharmaceuticals, where variables are measured more objectively, and less in areas such as opinion research.