When not to use the elastic net penalty in regression? There is no shortage of resources on the advantages of using an elastic net penalty over L1 and L2 penalties (LASSO and ridge respectively).
The following questions pertain to situations where regularization is desirable, i.e., looking for answers contrasting elastic net with other regularization strategies.
When doing regression analysis, when should one not apply an elastic net penalty?  Or more generally, what should one be wary of when choosing elastic net penalty over other regularization methods?

I am not aware of any practical situation where Ridge or Lasso are preferable to Elastic Net. The large-sample (asymptotic) theory for Ridge and Lasso seem to be better developed, so people may use them when they develop theory or if they want theoretical guarantees on the performance of their method.
As OP is already aware, Ridge and Lasso are a special case of Elastic Net. Elastic Net minimizes the function
$$ \hat{\beta} \equiv \underset{\beta}{\operatorname{argmin}}\left(\|y-X \beta\|^{2}+\lambda_{2}\|\beta\|^{2}+\lambda_{1}\|\beta\|_{1}\right).$$
When $\lambda_1 = 0$ and $\lambda_2 > 0$, Elastic Net becomes Ridge regression, and when $\lambda_2 = 0$ and $\lambda_1 >0$, it becomes Ridge regression.  Generally, we select the best values of $\lambda_1$ and $\lambda_2$ using cross validation. If Ridge or Lasso were to outperform Elastic Net in a particular case, cross validation would choose a $\lambda_1$ or $\lambda_2$ that reduces the  model to Ridge or Lasso.
In the special case when the design matrix is orthonormal, Zhou and Hastie give the closed form solution of each coefficient $\beta_i$ estimated by Elastic Net, Lasso, and Ridge:
$$\text{Elastic Net:} \enspace \hat{\beta}_{i} =\frac{\left(\mid \hat{\beta}_{i}(\text { OLS }) \mid-\lambda_{1} / 2\right)_{+}}{1+\lambda_{2}} \operatorname{sgn}\left\{\hat{\beta}_{i}(\text { OLS })\right\}$$
$$\text{Lasso:} \enspace \hat{\beta}_{i} \text { (lasso) }=\left(\mid \hat{\beta}_{i}(\text { OLS }) \mid-\lambda_{1} / 2\right)_{+} \operatorname{sgn}\left\{\hat{\beta}_{i}(\text { OLS })\right\}$$
$$\text{Ridge:} \enspace \hat{\boldsymbol{\beta}} \text { (ridge) }=\hat{\boldsymbol{\beta}} \text { (OLS) } /\left(1+\lambda_{2}\right)$$
Zhou and Hastie match up terms and plot the solution paths of each estimator to explain the behavior of Elastic Net. They state "elastic net can be viewed as a two-stage procedure: a ridge-type direct shrinkage followed by a lasso-type thresholding."  This further supports the idea of Elastic Net always outperforming Ridge/Lasso. If either direct shrinkage or thresholding isn't necessary, you can simply omit this step with Elastic Net
The classic example of Ridge outperforming Lasso is when you have many correlated predictors. but this does not appear to negatively impact Elastic Net. In the same paper, Zhou and Hastie examine the problem of correlated predictors analytically and through simulations. They find that the predictive performance of Elastic Net isn't negatively impacted in the same way Lasso is by correlated predictors.
The only benefit of Ridge and Lasso over Elastic Net I'm aware of is that they can be fit faster than Elastic Net. Ridge and Lasso only have a single tuning parameter while Elastic Net has two tuning parameters. However, Elastic Net can be fit quickly with existing software so this may not be a meaningful increase in computation time.
There may be a pathological situation where Elastic Net performs worse than  Ridge and Lasso. The only way this could happen is if something in the data causes a poor selection of $\lambda_1$ and $\lambda_2$.
EDIT: User @richard-hardy points out in a comment that Lasso/Ridge vs Elastic Net can be interpreted as a bias-variance tradeoff. The additional parameter in Elastic Net increases the variance of the model relative to Lasso and Ridge. That is, Elastic Net is more likely to overfit than Ridge or Lasso. Richard and I both suspect this isn't an issue in practice, and increased variance doesn't dominate the reduced bias of Elastic Net.
