We leave our oven on with door open to heat our trailer Will that cause the ovens thermostat go out it's an electric stove with four burners to boil water also for heat.
Have to start by saying, a much better idea to heat with an electric than with a gas stove: as mentioned in this excellent question, gas stoves emit toxic carbon monoxide.
Having said that, there are several major issues with heating with your electric oven, some of which this article from ehow.com points out:
Melting dials, etc. Ovens are not designed to be run open, hence to do so can set up the risk of melting plastic dials or other plastic parts. That being said, provided you take standard precautions, it's probably not an issue.
Breakdowns. This is obviously somewhat variable, but generally, I expect an oven to break down between 5-10 years, though one online article says you can expect 10-15 years. Obviously, the more you use the oven, the sooner it is likely to break. And needless to say, the cost of replacing or fixing a range is significantly higher than the cost of a space heater. You can easily get a decent space heater for well under a tenth of the price of an range. So even if your space heater only lasts 2-3 years and you need two of them to do the job of the oven, it comes out cheaper, plus giving a more even heat.
And I add two extras, which don't have to do with the oven itself, as your question, but as long as we're on the topic:
Cost. There is a price tag attached to heating with an electric oven: the oven was designed for even heating of a cooking substance, not heating the outside air. Thus, it outputs less heat per amount of energy burned than a simple space heater, which is designed to pump all the hot air out quickly and evenly.
Accidents. Obviously, if you ever have children around, an open oven at top temp is an invitation for disaster. Even for adults, a slip or a trip could result in burns. That being said, you could counter that issue by simply placing it in a safe place or erecting protective grill. This is obviously no worse than a wood or coal stove.
The control system will keep sending current through the heating coils, as the temperature will never reach the set value. With the door closed, the temperature will rise to its set level and the insulation will make sure that the coils only have to switch on intermittently to maintain a constant temperature. With the door open and air continuously circulating, the coils will be working the whole time that the oven is switched on. So theoretically yes - The overall life of your oven will decrease.