Thermometers are built to function within their range. That being said, they are also built to withstand temperatures slightly outside their range so that they are not damaged accidentally. So, if the temperature is within this range, no harm, no foul. Nothing happens. But in temperatures beyond that, they will break.
To understand this, we have to understand how a thermometer functions. In the thermometer ( the ones used commonly made of glass having a capillary for the rise and fall of fluid and a reservoir that holds the fluid when not in use) , there is a thermometric fluid that expands uniformly for every unit rise in temperature ( and contracts in a uniform manner for every unit fall in temperature as well) . So, when we use the thermometer beyond its range, the fluid expands until it has no more room to do so. But even now, at this temperature, it needs more room to expand. So, it makes more room by breaking the thermometer and flowing out.
The fluid used is usually mercury or alcohol. This is quite similar to how we keep blowing air into a balloon and after a certain point, if we try to blow further air, it bursts.