No, the order, via the lawful superior, is expected to not cause violations of other, extant vows and/or obligations. That said, they can require one to fulfill those obligations in a different rite.
For example, as a Byzantine, the good brother JR is expected to fulfill his Jan 6 Theophany obligation by attending a divine worship service. His superior needs to see to it he can, under pain of sin, as a superior… but neither he nor his superior are restricted to fulfilling that in the Byzantine Rite. Likewise, his order can add for him, by rule, custom, or instruction from his superior, an obligation for the Jan 1 feast’s liturgy, even tho’, as a byzantine, he’s not obligated; he can be obligated as a Carmelite.
Several orders require daily divine worship attendance. Those orders have to make it possible for their members to do so in a practical fashion, or dispense the obligation when not practical.
One of the interesting things about leadership in the Church is that it comes with responsibility for all the souls entrusted to one’s care, under pain of sin. If you don’t enable them to fulfill their obligations, you also suffer thesin. Hence the saying “The Road to hell is paved with Bishop’s souls.”