Commander used to have special rules about color identity and mana generation. As of January 18, 2016, these rules have been removed. Tapping lands for mana works just like regular Magic.
Note that the restriction on mana production and color identity in deckbuilding still applies. So your red/blue deck still can't include a Crumbling Necropolis.
You can, however, use all these exceptions to pay off-color costs:
Mana generators without mana symbols
Note that the deckbuilding restriction only applies to mana symbols, not "any method of generating other colors of mana."
The most common example is that cards that say "any color" aren't considered to have mana symbols in the card text. They may safely be included in non-five-color decks.
Borrowing your opponents' stuff
Say you're playing Ashling (red), and I'm playing Sen Triplets (white/blue/black). If I steal one of your Mountains (e.g. by playing it from your hand with my commander's special ability), I can tap it for red mana.
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. 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.)