Can motherboard heatsink screws change the inductance of nearby traces?
Does the screws affect the impedance of the traces?  Probably.  

Does it affect the trace impedance in any significant way?  Probably not.  

Do I think it is a major problem?  Absolutely not.  

Should you avoid that kind of heat sink?  I have never heard of a heat sink making a motherboard not work (due to impedance).  If a motherboard is that sensitive then I would avoid the motherboard, not the heatsink.

The influence on impedance is greatly dependent on distance, and once the distance is more than about 0.1 inches there is almost zero practical effect.  A distance of even 0.03+ inches is still quite minimal.