How does variance-inflation factor creep into a chunk test? In this question, the OP runs a “chunk test” and has a linear relationship between a variable in the restricted model and the variables in the chunk.
If this were a chunk of just one variable, that linear relationship would manifest through the variance-inflation factor and inflate the p-value of the t-test of that one variable. When we test a chunk of multiple variables, something similar should happen, but what is the math of how the variance-inflation factor appears in the F-test?
For one example of how this could happen, we could run an ANCOVA but have the factor variable being tested be somewhat predictive of our covariate. However, I am interested in more generality. The restricted model can have multiple variables, and the chunk could be any chunk of variables (not just a categorical variable like the ANCOVA example).
EDIT
EdM gave a nice answer to what I posted, but to clarify, my interest is more about relationships between in-chunk and out-of-chunk variables than it is about relationships within the chunk being tested.
 A: The variance-inflation factor for an individual predictor in an ordinary least-squares model is based on the multiple $R^2$ for its linear regression against all the other predictors.* Any correlation with the other predictors inflates the variance of its coefficient estimate beyond an uncorrelated situation.
The quadratic form involved in a simultaneous "chunk" test on multiple coefficients, in contrast, means that correlations between predictors can either increase or decrease the value of the test statistic depending on the signs of correlations and of coefficient estimates.
A simple example with two predictors illustrates the principle, with the additional simplification that both predictors have the same variance inflation factor. Start with the usual ordinary least squares setup, with continuous outcome $y$ and normally distributed error $\epsilon$:
$$y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \epsilon.$$
Center and scale each of $x_1$ and $x_2$ to zero mean and unit standard deviation. That simplifies the design matrix $X$ so that:
$$X^TX=N \begin{pmatrix}1 & 0 &0\\0&1&r\\0& r& 1 \end{pmatrix}$$
where $N$ is the number of observations and $r$ is the sample correlation coefficient between $x_1$ and $x_2$. The covariance matrix  of coefficient estimates, with $\hat\sigma^2$ the estimate of error variance, is:
$$\hat\sigma^2 (X^TX)^{-1} = \frac{\hat\sigma^2}{N} \begin{pmatrix}1 & 0 &0\\0&\frac{1}{1-r^2}&\frac{-r}{1-r^2}\\0& \frac{-r}{1-r^2}& \frac{1}{1-r^2} \end{pmatrix} .$$
The diagonal elements are the variances of the individual coefficient estimates, with variance inflation $1/(1-r^2)$. The covariance between $\hat\beta_1$ and $\hat\beta_2$ is opposite in sign to $r$. The Wald statistic for the hypothesis $\beta_1=0$ is then:
$$\frac{N \left( 1-r^2 \right) \hat\beta_1^2}{\hat\sigma^2},$$
with a corresponding result for $\beta_2$. Higher values of $r^2$ lower the values of test statistics on the individual coefficients, regardless of the sign of $r$. With more predictors, $1/(1-r^2)$ would be replaced by variance inflation factors in single-coefficient tests.
A Wald "chunk" test on the joint hypothesis that $\beta_1=\beta_2=0$ is based on the quadratic form between the vector $\begin{pmatrix}\hat\beta_1 & \hat\beta_2\end{pmatrix}$ and the inverse of the corresponding submatrix of the coefficient covariance matrix (last two columns and rows, in this example). The inverse of that submatrix is simply related to the corresponding submatrix of $X^TX$, giving:
$$\begin{pmatrix} \hat\beta_1 & \hat\beta_2\end{pmatrix} \frac{N}{\hat\sigma^2} \begin{pmatrix} 1 & r \\r&1 \end{pmatrix} \begin{pmatrix}\hat\beta_1 \\\hat\beta_2 \end{pmatrix}= \frac{N}{\hat\sigma^2} \left( \hat\beta_1^2 + 2 r \hat\beta_1 \hat\beta_2 + \hat\beta_2^2\right).$$
A positive correlation $r$ between $x_1$ and $x_2$, other things being equal, will increase the value of the "chunk" test statistic over the uncorrelated situation if $\hat\beta_1$ and $\hat\beta_2$ have the same sign, as will a negative correlation if the coefficient estimates have opposite signs.

*There's a generalized variance-inflation factor for other forms of regression; see this page.
