Dylan;

Precisely locate resistance in the current path, which causes heat, and correct, if possible. This is typically at transitions like riveted-on contacts, rivets to conductors, connectors to their mates, spring-loaded contacts, even wires in crimp-contacts. Once you have located problem, it may very well be repairable...

"Looking" around the suspect area with an IR thermometer might further help preciously localize the heat source...but it's important to not take too long to do this!...smoke obviously means a non-conductor is getting heated past some safe operating point and is carbonizing (and deteriorating rapidly), and after some amount of this, it will either mechanically fail and fall apart(!) or even ignite altogether. Work quickly...if necessary repeatedly with cool-down time between! Note also that circuit resistances drop voltage, so the resistance in the current path may also be located by voltage measurements, but this may not give a precise location...

I am somewhat surprised by Planetman (Eric's) explanation... the first part is absolutely correct and spot-on "corrosion increases the resistance", but I have to disagree with the second part "and causes a higher draw" ...there is no increase in current!...in-fact there is actually a decrease, since any additional Resistance in the current path will decrease total circuit current (Itotal = V/Rtotal). In this case, the situation is, that this (slightly to somewhat decreased) current, due to the additional Resistance, although apparently still being adequate to operate the Load as reported, is still way too much for some series connection location or junction..."which causes heat." (resulting in smoke).

High tech lighting circuits, like Xenon or HID, which have a "smart" power supply circuit between the power source and Load will draw more current when supply voltage drops, as the smart supply adjusts to keep the output power applied to the lights constant, but for simple lighting circuits, like were talking about here, circuit current is strictly determined by Ohms Law.

See also: http://www.sw-em.com/Fuses,%20Allocation%20and%20Troubleshooting.htm#Corrosion_Caused_a_Blown_Fuse

Good Hunting!