"The fan is not for cooling. It is a mosquito trap and the fan sucks mosquitoes attracted by a chemical compound, heat and UV light. The fan must not stop or the mosquitoes will escape. "

When it STARTED to run with a blown bulb what happened ? It ran. So it should run again. However, if it is sending already upped voltage to the CCFLs a regular switch is not reliable.

Anyway, it sounds like this contraption is a light, a heater and a fan. Common power supply ? That is common. Correct me if I am wrong.

If I am not wrong I say in the very long run you might cost a little longevity running without that part of the load, but I say MIGHT. Not for sure. And it's not like it's going to fail overnight.

Don't sweat it.