Heater fan causes fluorescent light to dim/flicker
This is indicative of overloaded circuits and/or improper wiring. Check the voltage at an outlet both with and without load. Recommended range for a nominal-120 VAC line is 110-124 VAC for Range A.

If the voltage drops below that, or if circuit breakers trip frequently, remove loads (especially a heater, which can draw 15 A) or rewire the house. Some older houses had aluminum wiring, which makes poor contact at connections with age.  Don't risk a fire caused by overheating circuits.