If I change my electric guitar tuning to Drop C (CGCFAD) from standard tuning (EADGBe), should I adjust the truss rod? Same question for other non-standard tunings.
If you intend to keep the guitar in that tuning permanently, it may indeed be a good idea to loosen the truss rod. The lower string tension will likely lead to the neck pulling back a bit, resulting in lower action – not something you want especially with those slack bass strings. But it depends on the instrument.
If you switch tuning for just a single song, then it's definitely not practical to also adjust the truss rod. Generally avoid doing that too often, it may wear out the mechanism or even lead to the neck developing a warp/twist.
You will only know when the tuning has settled. Since it's providing less tension on the neck, it won't harm the guitar, but when the action and intonation are sorted, you may find the strings rattle on the fretwires at some points. That's the time to adjust.
To keep that sort of tuning, it migt be a good idea to consider what gauge strings are needed, since the originals may be too floppy.