Control which features are used for every task in multioutput classification?
I don't think its possible. 

First, let's see what's the difference in explicitly modeling separate trees for different tasks versus modeling them in joint manner.

Lets suppose we have 2 task with each n classes. In the later case, to be able to jointly model the correlations, one must create new classes which is a subset of all the permutations available from nC2 combination of the classes from the 2 tasks. Now, if it is true that one feature (say feature A) is beneficial for task 1 but not for task 2, then how would one decide whether to use the feature A while figuring out the final classes in both the task? Task 1 wants the feature but Task 2 doesn't, so this creates a deadlock which would prevent us from putting a bias in the tree model to not use feature A for the classification! 

So, if you are certainly sure that a particular feature is not beneficial for a particular task the then the simpler and more effective way would be to model them separately as you mentioned.