iMac Core i3 mid 2010 hard drive removal and fan control
The usual result of removing the thermal sensor is roaring fans; I suspect that there's little reason for Apple to add some "if no drive, ignore sensor" logic, because Apple would want to replace the drive.

The best instructions for replacing components on your Mac can be found here:
https://www.ifixit.com/Search?query=imac%202010

Remember that your iMac has a SATA II (3 Gbps) controller, so you're not going to see the full benefit of the SSD: though it will probably double your HDD's speed.

If your iMac had Thunderbolt, then I would have said avoid the hassle and use an external drive; but it is a 10-year-old Mac!