I have trained my model with RandomForestRegressor, but now my training data is updated continuously. So I have to train my model with all the train data set i.e past and new data, or can I directly train my model only with new data? But if I train model only with new data, will it keep the pattern in existing data or not?
For sure you have to make sure your final model has seen all the history training dataset some time .
The only chance you have is to retrain your model (and not all the training algorithms accept it) in the fashion of incremental learning, where an already trained model is updated by being exposed to the new incoming data (used in online learning for instance), but it does not mean to completely retrain your model only on new data. In case you need it, have a look at it: https://scikit-learn.org/0.15/modules/scaling_strategies.html#incremental-learning where decision trees are not actually included for incremental learning
In your case, I recommend you (in case you can deal with performance cost) to retrain with all your available dataset.