Interaction terms and causal interpretation I'm estimating the following model:
$Y_{i,t} = \alpha_{0} + \alpha_{1}X^{1}_{i,t} + \alpha_{2}*T_{t} + \alpha_{3}X^{1}_{i,t}*T_{t} $
In which $T_{t}$ is the year treated as a continuous variable, tend to measure the linear trend of $Y_{i,t}$.
To give you some background, this work targets a field where using hypothesizing plus regression (not those like DID, RD, IV, etc.) analysis is still comfortable for researchers to infer causality in certain ways. But they still care about the standard issues (e.g., multicollinearity, omitted variable; measurement error, etc.) in interpreting results from the causal view.
Anyway, in this particular project of ours, we have the following hypotheses:
1, $X^{1}_{i,t}$ affect $Y_{i,t}$ .
2, the effect of $X^{1}_{i,t}$ on $Y_{i,t}$ changes linearly with time (i.e., year).
The model above is to test hypothesis 2. Therefore, we expect $\alpha_{3}$ to be statistically significant. And that is actually what we got.
But to interpret $\alpha_{3}$ in a causal sense, someone raised the concern that the effect of the interaction term may come from $T_{t}$ if it is correlated with $X^{1}_{i,t}*T_{t}$.
Thanks to the answers, I realized that this is a multicollinearity issue.
First, there is multicollinearity between interaction term ($X^{1}_{i,t}*T_{t}$) and variables ($X^{1}_{i,t}$,  $T_{t}$) constitute it.
Second, there may be multicollinearity between $X^{1}_{i,t}*T_{t}$ and  $T_{t}$ arise from the data generating process.
The first one can be taken good care by centering before generating the interaction term. But what if there is still multicollinearity between $X^{1}_{i,t}*T_{t}$ and  $T_{t}$ even after the centering procedure?
1, How can we test whether the second type of multicollinearity exists?
2, How to deal with it if it does emerge from the result of the test?
 A: Your model is a regression with a (linear) time component and a time-treatment interaction. Otherwise you explain nothing about the problem or the data you are working with. In this very general setting, you cannot interpret regression coefficients (either main effects or interactions) causally without further assumptions about the data generating process.
The reason: correlation is not causation as you probably already know. Regression tells us something about the relationships between the outcome Y and the predictors X and T (ie, about correlations). Extra conditions are required to interpret those relationships as causal. In other words, it's true that if X causes Y and its causal effect increases/decreases with time, then the interaction term is non-zero. But the interaction term is non-zero under a variety of other situations where X doesn't cause Y.
Since there is no generic argument that regression reveals causal effects, you need to focus on your specific problem/data/question to demonstrate causality.
(Free) Resources to learn about causal inference: 
[1] Causal Inference for The Brave and True causality according to a data scientist 
[2] Causal Inference: The Mixtape causality according to an economist 
[3] The Effect: An Introduction to Research Design and Causality causality according to an economist, v2 
[4] Causal Inference: What If causality according to epidemiologists 
[5] Introduction to Causal Inference from a Machine Learning Perspective causality according to an ML scientist 
A: You should accept the answer from @dipetkov (+1), as the difficulties of causal interpretation are key.
In terms of statistical analysis per se, note that $T_{t}$ is necessarily correlated with $X^{1}_{i,t}*T_{t}$; consider the perfect correlation if $X^{1}_{i,t}$ is constant with time. That's inherent in an interaction term.
The statistical model addresses whether including the $X^{1}_{i,t}*T_{t}$ term improves the model with just $X^{1}_{i,t}$ and $T_{t}$ as separate predictors. A significant coefficient for the interaction doesn't just gauge whether "the effect of $X^{1}_{i,t}$ on $Y_{i,t}$ changes linearly with time"; it also indicates that the effect of $T_{t}$ on $Y_{i,t}$ depends on the values of $X^{1}_{i,t}$. Those might have different "causal" interpretations.
And furthermore, with this type of time-series data a spurious correlation can wreak havoc with causal interpretation.
