Let's use the power formula.
So in the circuit:
The total current through the circuit is
P = (12^2)/220
P = 0.654545 Watts
Now all you need to do is check whether the 220Ω resistor and the 10KΩ Potentiometer can both handle the wattage (
0.654545 Watts). As you said, the 220Ω resistor is
0.25 Watts rated so that will definitely blow up. If the potentiometer is not rated for that Wattage, they both blow up.
Now let's calculate what resistance you need to replace the 220Ω resistor. If the potentiometer has a higher wattage rating than the 220Ω resistor, it makes no difference.
I am going to assume the Potentiometer has a wattage rating of 0.1 W
Since the Potentiometer is 10k, max current is 3mA due to
I = root(P/R)
I = root(0.1/10000)
I = 0.003162A
We are going to replace the 220Ω with something better. Assuming your other resistors are rated 1/4 Watts, let's find the minimum required resistance at 12V
R = (V^2)/P
R = (12^2)/(1/4)
R = 144*4
R = 576Ω
That's the minimum resistance for the resistor. Let's see about the potentiometer.
R = V/I
R = 12/0.003162
R = 3794Ω
Jeez a high value...
Now although the resistor only requires about 600Ω for itself, the potentiometer requires a whopping 4K resistor, so you have to choose the higher one.
Now we don't want our resistor to be on the plain boundary of the maximum Wattage rating, so get like a 4KΩ resistor or more (1/4 watt rated). Also pay attention to the tolerance. For eg if the tolerance was 10%, then the resistor could be 3600Ω which is very dangerous.