Would the climate still rise when we reduce our carbon footprint down to zero?
While I can't comment on whether the global temperature would still rise if humanity's carbon footprint were reduced to zero, I do know that the heat our machines produce would not contribute any net warming effect. The issue is not the amount of heat which is entering the atmosphere -- it's where and when that heat came from.
Assuming that we are successful in reaching 100% wind, water, and solar energy, the energy consumed at the end use would be from only seconds to at most a year ago:

Seconds: Solar power from photovoltaic panels is collected, converted to electricity, transmitted to the grid, and used almost instantaneously
Hours to days: Wind is primarily generated by daily fluctuations in surface temperature caused by the sun
Months: The water in rivers which power hydroelectric dams is generated by seasonal patterns in precipitation, which is a function of changes in solar intensity over the course of the year

In contrast, the energy we're using today from fossil fuels is tens to hundreds of millions of years old, containing energy harvested from the sun over a period of over a hundred million years (source).
The energy from wind, water, and solar sources is being added to our atmosphere anyway -- we just convert it into different forms to use it for other purposes in between its transit from the sun to the planet.
By contrast, the energy from fossil fuels had been locked away, safely stored for 100 million years or more. Until we started drilling and mining, it was no longer having any significant impact on the planet.
This also applies to the food we consume -- the energy that fruits and vegetables store is at most a year old -- for animal products, it's at most two to three years old.