Is there any way to put iMac to sleep (or other low power mode) while using it in target display mode? When I connect to my mid-2010 27" iMac in Target Display Mode, fans hum and hard disks spin (with occasional I/O). Being logged out in iMac's macOS doesn't help.

Is it possible to lower this unnecessary power drag or at least minimize the noise? Can I boot iMac as a dumb-display?
No, you cannot go into low power mode while in Target Display mode.

While your iMac is "acting" as a "passthrough" for you display signals you are not bypassing the logic board and GPU to connect to the display.  The iMac display is still connected to the internal logic board and GPU and that GPU is required to display images on the screen.

The fans are coming on because the SMC in the iMac has calculated (based on input from the temp sensors) that it has reached a certain temperature threshold and it needs to cool down.  Remember, your GPU and logic board is hard at work taking DispayPort and/or Thunderbolt input and converting it to displayable graphics.

If you want a "dumb" monitor, the best option is to get a dumb monitor.