I accidentally smoked a potentiometer with an Arduino Uno. I plugged in the 5V and analog signal wires the wrong way round and the pot started smoking, so I quickly unplugged it, but potentiometer is probably destroyed. But is there a chance the pot still works and could have I caused any damage to the board?
It is near impossible to know if the board is damaged. It sounds like you connected 5v to the middle pin. Remove all pin connections and try reconnecting it with a simple program that attempts to use that pin. Maybe start with the example blink sketch and circuit on that pin. If it works you are OK. The regulator would be more likely damaged than the chip from the sounds of things.
Potentometer is probably damaged to be honest - even if it still shows the correct resistance on a multimeter. I'd also be concerned about anything else connected to the potentiometer.
Why ask us? I mean you have the pot and the Arduino, you are the only one who can test them!
You might be unlucky and the tracks burnt out. You might be lucky - the track is carbon, so can handle
highish temperature. The track might be scratchy but work, it might start to break up mechanically.
The Arduino's regulator is the only component that got strained and it ought to be able to thermally
cut out and protect itself, so I'd expect the Arduino to work fine.
I agree... The Arduino is probably fine... Just try downloading & running the Blink sketch.
The pot is probably damaged, and maybe "totally fried". You can check the pot by connecting it correctly and running the Analog Read Serial sketch. Or, you can just throw it away. (Pots aren't expensive, but you wouldn't want to order just a pot because of the shipping costs.) Even if it still works, I'd guess it's got some "bad spots" somewhere in the rotation (as Mark put it "scratchy" if you were to use it as a volume control).
So... Now is a good time to think about getting a multimeter! (I assume you don't have one?) You can check the pot with a multimeter, and more importantly check the connections next-time before you wire it up! If electronics is your hobby you need a multimeter, and if you have limited funds a cheap meter is better than no meter.