Pressure is not additive
If you mix two gases of different pressure, the resulting gas does not have a pressure equal to the sum of the two pressures. Treating the gases as ideal gases, the pressure will be the average of the two pressures, weighted by the volumes. What is additive is the partial pressure of a gas mixtures, where the sum is the total pressure.
Closed container with constant volume
To simplify things, let's place a closed bottle filled with the liquid of interest into the container, adjust the pressure to 1 atm, and close the container. If we heat the container, the pressure will go up just because the gas temperature rises. If we open the bottle, some of the liquid will evaporate until the partial pressure is equal to the vapor pressure. Again, the container will have a pressure higher than atmospheric pressure. So the liquid will not boil when the temperature reaches the normal boiling point because the pressure will be above atmospheric pressure at that point, for the two reasons mentioned.
Closed container with variable volume
If the experiment is done in a closed, variable volume container that expands into the surrounding kept at 1 atm, the volume will increase as the temperature is raised (because the gas expands, and because the vapor pressure increases with increasing temperature). When the temperature reaches boiling temperature (slightly above, actually), the liquid will boil until it is all gone, and the volume will increase further during this boiling process. The partial pressure of the liquid will never reach 1 atm (because there are other gases present), so the vapor pressure will remain higher than the partial pressure throughout the boiling process.
Does the total pressure become twice the atmospheric pressure at the surface?
No. Either there is room to expand, and the gas produced by boiling will push away the other gas (and then quickly mix with it), or we keep the volume constant, and then we don't call it boiling (because the total pressure exceeds 1 atm, so the liquid will not boil. However, when you heat a liquid to the boiling point and let the system reach equilibrium, the partial pressure of that substance will be 1 atm, and if any other gases are present, the total pressure will be higher. How much higher depends on the starting conditions. If our starting condition is a completely filled closed bottle, with inert gas (e.g. nitrogen or air) already at the boiling point of the liquid and at 1 atm, when we open the bottle the total pressure will reach 2 atm.