Is the exhaust fan necessary?

It is worth checking if your BIOS has options to shut down the PC based on the temperature of the CPU. This means if you do decide to run without the exhaust fan then your PC would shut itself down if things were getting too hot preventing damage to the components.

And in reply to your bracketed comment, I would say it's really worth looking into replacing the case fan. The difference in noise between stock components and third-party replacements costing only a few dollars can be huge. Another option would be a fan speed controller so you can lower the speed of the case fan so that it is much quieter but still provides some air flow.

This is a good article on airflow in a computer case.

If the PSU fan already seems to provide enough air-flow, the additional case fan can be left off. Even more so as though your graphics card has passive cooling.

It looks like the additional fan is overkill in your case.

If you find that you still need it, try to find vibration-decoupling fan mounts. They help a lot.