It's impossible to add off-color mana to your mana pool. That means that, short of replacement effects that alter how mana can be spent, it's impossible to pay off-color mana costs.
So, what do those replacement effects look like?
Changing mana's function without changing its literal color
Cards like Mycosynth Lattice allow you to spend mana "as if it were any color of mana" without actually changing the colors of that mana.
There's a really technical caveat to that, though...
I'm pretty sure that, even with Mycosynth Lattice in play, there are still corner cases where mana colors matter: for instance, Mycosynth Lattice will allow me to spend 5 blue to cast Torrent of Souls, but the spell still checks the color of mana spent on it (which, in this case, causes it to have no effect). This is based on the rulings text of Celestial Dawn:
You may use a different color mana than the color required for spells
and abilities that require a specific color. If you do, and the spell
or ability checks the actual color of the mana, it can tell the
Alternatives to mana
Some rules allow you to "pay for" mana costs without using mana at all. Since this bypasses your mana pool, you can use it to pay off-color costs. For example:
(Note that color identity still applies: you can't put Chord of Calling in your Sen Triplets deck. But, if you've got some green creatures, you can cast someone else's.)