What causes lights to get brighter when furnace motor comes on?

A loose or disconnected neutral wire at breaker box or at the appliance. Depending on what is on when and if two or more motors are running at the same time. This can as well have the opposite effect too, lights too bright.

Although loose connection can cause a problem and you should have all the connections checked, it could also be caused by several other reasons. You distance from and the load on the transformer, the size of you service conductors or lateral, the age and maintenance of your appliances. I have found out that light dimming happens in almost every house but some people are more sensitive to it than others.

Usually dimming of lights when an appliance kicks it is caused by a voltage drop of the supply caused by a heavy demand for current when the appliance initially starts. This is usually associated with an appliance that has a motor such as a refrigerator, furnace, washing machine, vacuum cleaner etc. Motors demand about 6 times their steady running current when they first start up and this can be a significant surge of current for maybe a half second or so till the motor has started and is up to speed. This basically momentarily 'overloads' the supply and lowers the voltage available to other devices, thereby causing them to momentarily slow down (if a motor) or dim (if it is a lamp).

If you switch to the new power saving compact spiral bulbs, their internal electronic ballast compensates somewhat for line voltage, so they won't react as noticeably to heavy loads starting up elsewhere in your house. The same is true for fluorescent tube fixtures with electronic ballasts.

One more point:If the lights are on a dimmer, they are very sensitive to change in voltage and will change their brightness disproportionately to the slight changes in voltage caused by motors or other heavier loads switching off and on.