Is hyperparameter tuning more affected by the input data, or by the task?
In other words, is hyperparameter tuning more affected by the task (which is constant) or by the input data?


It's correct that the task is constant, but hyper-parameters are usually considered specific to a particular learning algorithm, or to a method in general. In a broad sense the method may include what type of algorithm, its hyper-parameters, which features are used (in your case which embeddings), etc. 

The performance depends on both the data and the method (in a broad sense), and since hyper-parameters are parts of the method, there's no guarantee that the optimal parameters stay the same when any part of the method is changed even if the data doesn't change.

So for optimal results it's better to tune the hyper-parameters for every possible pair of ML model and words embeddings. You can confirm this experimentally: it's very likely that the selected hyper-parameters will be different when you change any part of the method.