Why is interpolation a time varying system
Considering a discrete time framework, the interpolation operation has two stages; first expand the signal by zero stuffing between the existing samples, and then lowpass filter the expanded signal to get the interpolated samples. The second operation is LTI, hence it's the first stage which is not LTI.

The expansion block is shown like $$x[n]\longrightarrow \boxed{ \uparrow L } \longrightarrow y[n]$$ and mathematically defined as

$$ y[n] = \begin{cases}{ x[\frac{n}{L}] = x[m] ~~~ , ~~~ \text{ for } n = m L~~ ,~~  m=0,\pm 1, \pm 2,... \\  ~~~~~ ~~~~ ~~~ 0  ~~~~~ ~~~~ ~ , ~~~ \text{ otherwise } }\end{cases} $$

Now apply the test for time-invariance  on $y[n]$ :

First shift the ipnut. Let $x_d[n] = x[n-d]$, then call $y[n]$ as $y_2[n]$ as:

$$ y_2[n] = \begin{cases}{ x_d[\frac{n}{L}] = x_d[m] ~~~ , ~~~ \text{ for } n = m L~~ ,~~  m=0,\pm 1, \pm 2,... \\  ~~~~~ ~~~~ ~~~ 0  ~~~ ~~~~~~~~~~~~ ~, ~~~ \text{ otherwise }}\end{cases} $$

which becomes:
$$ y_2[n] = \begin{cases}{ x[\frac{n}{L} - d]  = x[m-d] ~~~ , ~~~ \text{ for } n = m L~~ ,~~  m=0,\pm 1, \pm 2,... \\  ~~~~~ ~~~~ ~~~ 0   ~~~ ~~~~~~~~~~~~~~~~~ ~ , ~~~ \text{ otherwise } }\end{cases} $$

Second, similarly shift the output $y[n]$ by $d$ which is denoted as $y_d[n] = y[n-d]$:

$$ y_d[n] = y[n-d] = \begin{cases}{ x[\frac{n-d}{L}]=x[m] ~~~ , ~~~ \text{ for } n-d = m L~~ ,~~  m=0,\pm 1, \pm 2,... \\  ~~~~~ ~ ~~~~ ~~~~ 0   ~~~ ~~~~~~~~~~ ~, ~~~ \text{ otherwise }  }\end{cases} $$

which becomes:
$$ y_d[n] = \begin{cases}{ x[\frac{n-d}{L}] = x[m] ~~~ , ~~~ \text{ for } n = m L + d~~ ,~~  m=0,\pm 1, \pm 2,... \\  ~~~~~ ~~~~ ~~~ 0  ~~~ ~~~~~~~~~~ ~, ~~~ \text{ otherwise } }\end{cases} $$

As you can see, $y_2[n]$ and $y_d[n]$ are not the same signals. You can explicitly see this by evaluating them for specific values of $n$ and $d$, such as $d=1$ and $n = L$, then $y_d[L] = 0$ while $y_2[L] = x[0]$ and hence $y_d[L] \neq y_2[L]$. And therefore the conclusion that the expansion system (or as a consequence the interpolation system) is a Time-Varying and not LTI system.