iMac late2009 fan doesn't shut down OS X10.11.6 (15G22010)

The fans run continuously. They only shut down when you turn off the machine. If the fan is running too loudly then there s a problem, not an annoyance.

Use Activity Monitor to Kill Runaway Processes

Use Activity Monitor on your Mac

Runaway applications can shorten battery runtime, affect performance, and increase heat and fan activity.

Intel-based Macs- Using Apple Hardware Test

upekkha-AppleHardwareTest · GitHub

Using Apple Hardware Test - Apple Support

A Troubleshooting Procedure that may Fix Problems With Lion to Yosemite

You should try each, one at a time, then test to see if the problem is fixed before going on to the next.

Be sure to back up your files before proceeding if possible.

1. Shut down the computer. Disconnect any third-party peripherals.
2. Reset your Mac's PRAM and NVRAM
3. Intel-based Macs: Resetting the System Management Controller (SMC)
4. Start the computer in Safe Mode. Test in safe mode to see if the problem persists, then restart normally. This is slower than a standard startup.
5. Repair the disk by booting from the Recovery HD. Immediately after the chime hold down the Command and R keys until the Utility Menu appears. Choose Disk Utility and click on the Continue button. After Disk Utility loads select the hard drive entry (mfgr.'s ID and drive size) from the side list. In the Disk Utility status area, you will see an entry for the S.M.A.R.T. status of the hard drive. If it does not say "Verified" then the hard drive is failing or failed. (S.M.A.R.T status is not reported for external drives.) If the drive is "Verified" then select the indented (usually, Macintosh HD) volume from the list. Click on the First Aid tab, then click on the Repair Disk button. When it finishes then click on the Repair Permissions button. When that has completed quit Disk Utility and return to the Utility Menu. Select Restart from the Apple menu.
6. Run the Apple Hardware Test. Use Apple Hardware Test and How to invoke and interpret the Apple hardware tests - CNET.
7. Repair permissions on the Home folder: Resolve issues caused by changing the permissions of items in your home folder.
8. Create a New User Account Open Users & Groups preferences. Click on the lock icon and enter your Admin password when prompted. On the left under Current User click on the Add [+] button below Login Options. Setup a new Admin user account. Upon completion log out of your current account then log into the new account. If your problems cease, then consider switching to the new account and transferring your files to it - Transferring files from one User Account to another.
9. Reinstall OS X by booting from the Recovery HD using the Command and R keys. When the Utility Menu appears select Reinstall OS X then click on the Continue button.
10. Erase and Install OS X Restart the computer. Immediately after the chime hold down the CommandandRkeys until the Apple logo appears. When the Utility Menu appears:

1. Select Disk Utility from the Utility Menu and click on Continue button.
2. When Disk Utility loads select the drive (out-dented entry) from the Device list.
3. Click on the Partition tab in Disk Utility's main window toolbar.
4. Set the number of partitions to one (1) from the Partition Scheme dropdown menu.
5. Set the Format type to Mac OS Extended (Journaled.)
6. Click on the Apply button.
7. Quit Disk Utility and return to the Utility Menu
8. Select Restart from the Apple Menu.

11. If none of the above helps then see How to Downgrade macOS High Sierra and macOS Reversion- How to Downgrade from High Sierra.

12. If you get here without success then it's time to make an appointment at the Apple Genius Bar for service. If you need to find an Apple Store - Find a Store - Apple.

Have you replaced the disk drive in this iMac?

I have a Late 2009 27" iMac and that year the disk drives Apple installed included custom Apple temperature firmware. If the disk was replaced, the iMac fans would ramp up to full speed. For a few years after replacing my hard disk, I used an app called "SSD Fan Control" (it was free) to regulate my fan speed.

Eventually, I got a special SATA disk cable from <http://MacSales.com> which makes the iMac think it has one of the original disks with the custom firmware.

If you still have the original Apple installed disk, then consider this information you can use in the future should you wish to replace the slow rotating hard disk with an SSD. For me putting an SSD in my iMac has kept it a fast and very usable Mac I used everyday at work.

NOTE: I do not sleep my iMac so I cannot tell you if the fan continue to run when it is asleep.