Generally no. Heat on earth is radiated out to space quite quickly.
When Mount St Helens exploded, it released about 1.0e11 MJ of energy. The earth receives 21MJ of energy from the sun, per day, per square meter. So MSH in total released as much energy in total as a square of land 70km on a side receives in a single day from the sun. Which isn't tiny, but is a relative drop in the bucket compared to the whole planet.
Considering that this hypothetical heat source would be very concentrated and so would radiate to space efficiently and not spread out as much, I wouldn't expect it to have much of any effect on the climate.
When Mount St Helens exploded, it released about 1.0e11 MJ of energy. The earth receives 21MJ of energy from the sun, per day, per square meter. So MSH in total released as much energy in total as a square of land 70km on a side receives in a single day from the sun. Which isn't tiny, but is a relative drop in the bucket compared to the whole planet.
Considering that this hypothetical heat source would be very concentrated and so would radiate to space efficiently and not spread out as much, I wouldn't expect it to have much of any effect on the climate.