Q:

Can we safely consume ions?

When we drink salt water, we are consuming the sodium and chloride ions, right? We can't consume elemental sodium as it's highly reactive but the consumption of its ions seems to be fine. Does that mean that we can consume ions of all elements?

A:

Nope:

Lead
Arsenic
Cadmium
Mercury (H/T @orthocresol)

And multiple others...

To clarify a bit further, the hazards from sodium metal and from, say, ionic mercury are different:
Sodium metal is so "eager" to cast away an electron that it will do so quite aggressively when it contacts any aqueous solution, including saliva.  It thus represents a "reactivity" or "chemical" hazard in the mouth.  Once each $\ce{Na^0}$ has forcibly donated its electron, though, it becomes the non-hazardous $\ce{Na+}$.
Ionic mercury, on the other hand, represents a relatively minor "direct" hazard to the body.  Instead, the various forms of mercury interfere with biological/biochemical processes in the body, leading to its severe toxicity.  One particular example highlighted in the Wikipedia article on mercury poisoning is the irreversible inhibition of selenoenzymes:

One mechanism of mercury toxicity involves its irreversible inhibition of selenoenzymes, such as thioredoxin reductase .... Although it has many functions, thioredoxin reductase restores vitamins C and E, as well as a number of other important antioxidant molecules, back into their reduced forms, enabling them to counteract oxidative damage. Since the rate of oxygen consumption is particularly high in brain tissues, production of reactive oxygen species is accentuated in these vital cells, making them particularly vulnerable to oxidative damage and especially dependent upon the antioxidant protection provided by selenoenzymes. High mercury exposures deplete the amount of cellular selenium available for the biosynthesis of thioredoxin reductase and other selenoenzymes that prevent and reverse oxidative damage, which, if the depletion is severe and long lasting, results in brain cell dysfunctions that can ultimately cause death.

