when I turn take key out car temperature guage raises and the on 2008 Chevrolet Equinox

engine fan runs

If I am reading this question correctly, you say that the Temp guage goes up and the fans come on AFTER you remove the key? Are there any codes stored in the BCM or PCM/ECM?
Not that I am aware of
Have you (or anyone else) scanned it for codes to see? You need to start there. If there are no codes, then you have an electrical short in the fan wiring circuit (includes relays), or internally in the BCM or PCM. The fans have 3 different relays and they are all controlled by the modules. You really need a scanner (not just a code reader) to diagnose the issue. You need to see if the fans are actually being commanded on by a module or if there is an external wiring issue.