Can a Guildgate be tapped for colorless?
You can use a guildgate to help pay for Karn, but it doesn't involve colorless mana.

When you see a grey circle with a number mana symbol in a cost, like the {7} in Karn's casting cost, that's actually called a generic mana cost. You can spend any color of mana to pay for that cost. So you can, say, tap an Izzet Guildgate for red (or blue) mana and use that to pay for one of that seven mana.

Colorless mana, on the other hand, is what's produced when you see something like "add {1} to your mana pool", again with the grey circle with a number, for example on Tomb of the Spirit Dragon. You can use that to pay for generic mana costs too - so you could use colorless mana to cast Karn, but you don't have to. Any color will do.