It is pretty useful information to know if a resistor will burn out just by the amount of voltage present within a circuit. Using the information from How to Determine Wattage you can calculate the maximum voltage rating for any resistor. The first value to determine is the maximum current:
Resistors always have a wattage rating and obviously have a nominal resistance value. These can be used to calculate a maximum current. This does mean that there are two solutions each time (since there is a square root involved because the current is squared). Just use the positive value, the negative value doesn’t make any sense in this context, they will both be the same value anyway. The second equation for power helps determine the max voltage:
Substitute the same power rating for the part and the value for current just calculated and solve for the voltage. For example, lets pick a random value and wattage rating for example, 150 Ohms with a 1/2 Watt rating. Here are through hole options that would be applicable: https://www.digikey.com/short/p0z4vj ; of course I am not including tolerance on resistance, so technically there will be a plus and minus tolerance on the voltage as well.
Solving for IMax will give +0.058A(58mA) and -0.058A(-58mA) (we only need the positive value). Substitute this value in the other power equation:
The final value for the Voltage would be roughly 8.621V. This makes sense because if you apply Ohms law and try 8.8V for example the current would be 0.0587A or 58.7mA which exceeds the maximum and the power required would be 0.517 Watts. Yes this is pretty close to the 1/2W rating, but any amount over the max wattage will burn out the resistor much faster than being under the max wattage rating. This is especially true if you go much over 8.621V, it will fail much faster the higher you go.