Is this a really low end motherboard? Are all your case standoffs installed properly?
This will cause early trace failures for sure, possibly other things. I would suggest that if the cooler is this heavy you probably need to consider putting something far lighter, or stock cooler if your CPU can operate that way. Lots guessing without knowing what this hardware is.
If you can't change the base hardware, like a thicker PCB mobo, an AIO cooler plate might be light enough to eliminate the board sag.
Early on in my PC building and modding I had come across a really nice ThermalTake tower cooler that was entirely copper and heavy as could be. In retrospect, completely unnecessary for the builds I was messing with. I ended up putting it on either an H or B chipset value board and not long started having strange issues with USB, memory, other assorted glitches and BSOD until one day it just didn't work. It wasn't until much later that I gained the knowledge to realize what had actually happened.
That cooler is actually still in use at an acquaintance's house on an ancient AMD.