The engine oil temperature (EOT) sensor is a thermistor. It changes its resistance based on oil temperature. The engine computer uses this data to adjust fuel delivery and variable valve timing. It is especially important in diesel engines and turbocharged vehicles to prevent overheating.
A failed EOT sensor will throw off cold-start fuel calibration. The engine might run rich and idle poorly. On diesel engines, it can cause the glow plug system to run incorrectly, making cold starts difficult.