Important features for the XGboost algorithm are also the most important for the training of DNN? I know that both a deep neural network (DDN) and the gradient boosting decision tree algorithm Xgboost can be used for the task of classification.
I'm using a DNN first and it works fine. With XGboost algorithm you can also discover which features are most important in the classification with the F test score. 

Can I state that the most important features for the XGboost algorithm are also the most important for the training of the DNN?

 A: The importance of features is a data-based thing. It does not directly depend on the model you prepared. However, some "importance" can be skipped or ignored by the model due to his nature. If your xgboost model recognized some features as important you should expect this feature would be important for other models. However if the nature of your DNN model is much different from xgboost, i.e. its convolution or recurrent, attention-based, you should NOT assume that feature that was not important for the xgb would not be important for DNN. You can expect that the transfer of feature importance between models will be more accurate if your models are similar in the analytical way, with a strong emphasis on the dynamic or static nature of model dichotomy and topology-based dichotomy.
A: Complementary podludek's nice answer (+1).
We cannot state that the most important features for the XGboost algorithm are also the most important for the training of the DNN. This is not only because our models might be dissimilar in terms of their structure (e.g. a gradient boosting model vs. a convolutional neural network) but also because: 1. they might be dissimilar in terms of the metric they optimise against (e.g. cross-entropy vs AUC) and 2. the feature importance method used might encapsulate different dynamics (e.g. SHAP values vs permutation importance).
Finally, I would draw attention to why the task of assigning feature importance is performed to begin with. If we do it so we can just focus on certain variables, that is mostly likely problematic approach. Overall model performance can be harmed when focusing exclusively on "important features". Using methods that employ regularisation within their fitting procedure (e.g. like Gradient Boosting Machines) is preferable to the complete exclusion of certain variables to the expense of others. On the other hand, if it is done as a sanity check to identify potential data leakage issues, or to recognise potentially expensive to acquire but low-contributing variables, or present to a layman audience the main drivers behind a model; it is a reasonable thing to do. 
