The following goes mainly for the acoustic guitars but the electric guitar is made of wood, too. So changes in temperature and humidity have effect on all guitars. This topic is important both in summer and in winter but as it is winter now, I will talk about the specific problems of winter time.
Ideally, the guitar wants to be in the same humidity where it was built. It would be ridiculous to try to keep it even all the time. But there are some limits that everyone should keep in mind: 40-50% of relative humidity in the air is the best for guitars. Some luthiers say that if a guitar is kept below 35% for any time then it will be damaged for sure. On the other hand, humidity over 60% isn’t good either.
Problems caused by dryness are more important. I live in a place where the air temperature can go as low as -20 C (occasionally more). It means that people heat their houses and it dries the air. So what can happen?
If the wood dries, it shrinks and many places get very tense. Sometimes the soundboard can crack. String action goes very low, glue can surrender somewhere. Everything shrinks but the metal. That means that the frets remain the same but you will start feeling their ends. If that happens then this is the last call to do something to save your guitar. You can use a special air humidifier or keep your instrument its case and put small humidifier in the soundhole. The simple solution is to put plastic bottles with water behind your radiators. Cut their neck so the water can get out more easily.
Most often you will have problems with solid wood instruments. Laminated wood is more stable. Of the two main materials of a soundboard the cedar is more risky than spruce. A crack in the soundboard is not the end of the guitar but there is no need to let things that far. By the way, most of the problems that are caused by the lack of humidity will not go away when the air normalises. See the picture above if you don’t believe. This is a living example from the last winter. Also, it should remind us why we don’t put metal strings to our classical guitars.