A Command Tower can produce any color of mana in its controller's commander's color identity. So, if your opponent controls one, your Exotic Orchard can produce any color of mana in that player's commander's color identity (assuming there are no other lands).
The relevant rule here is 106.7:
Some abilities produce mana based on the type of mana another permanent or permanents “could produce.” The type of mana a permanent could produce at any time includes any type of mana that an ability of that permanent would produce if the ability were to resolve at that time, taking into account any applicable replacement effects in any possible order. Ignore whether any costs of the ability could or could not be paid. If that permanent wouldn’t produce any mana under these conditions, or no type of mana can be defined this way, there’s no type of mana it could produce.
If mana outside the color identity of your Commander is ever added to your mana pool, that mana is considered colorless. So, in effect, it could only produce colors that overlapped between your and your opponents' commanders.