What makes heavy metals toxic?
Heavy metal toxicity is actually due to the inhibition of the body's natural enzymes as well as the denaturing of proteins that it causes. 

Here, Tchounwou et. al go into detail about the mechanics of different heavy metals and how they interact with the human body specifically. It even suggests ways that the heavy metals could be introduced. 

A short summary on the damage caused by what was studied:


Arsenic inhibits DNA repair. 
Cadmium toxicity isn't well known, but they think it causes single
strand DNA damage.
Chromium causes chromosomal abnormalities thought to be caused by
oxidative damage.
Lead prevents the body from interacting with Calcium which is a key
element in many biological processes. It also competes with other
metals for the active sites in different enzymes.
Mercury depletes the body's antioxidants.


So to answer your question, no, the heavy metals are not all working on the same mechanism. It all depends on what dose, how it was introduced, and the particular metal. Also, anything can be toxic in the right amount (people have died from water toxicity in drinking contests).