The first thing that came to my mind was iron. Human, and mammalian blood in general, 'captures' more oxygen than the normal oxidation process of iron. This process is complex, and requires many stages and components (enzymes, for instance). The blood system, and red blood cells (RBCs) in particular, have developed unique defensive mechanisms to preserve this functionality.
Studies from our laboratory have shown that oxidative stress plays a
significant role in damaging the RBC membrane and impairing its
deformability. RBCs are continuously exposed to both endogenous and
exogenous sources of reactive oxygen species (ROS) like superoxide and
hydrogen peroxide (H2O2). The bulk of the ROS are neutralized by the
RBC antioxidant system consisting of both non-enzymatic and enzymatic
antioxidants including catalase, glutathione peroxidase and
peroxiredoxin-2. However, the autoxidation of hemoglobin (Hb) bound to
the membrane is relatively inaccessible to the predominantly cytosolic
RBC antioxidant system. This inaccessibility becomes more pronounced
under hypoxic conditions when Hb is partially oxygenated, resulting in
an increased rate of autoxidation and increased affinity for the RBC
From Red blood cell oxidative stress impairs oxygen delivery and induces red blood cell aging for instance.
So the alien species, having evolved an entirely different blood system, would have a completely different method to preserve tissue integrity. If, for instance, the alien method required ROS (from the quote) systems to function properly, our blood antioxidant defensive system is designed to naturally neutralize such systems. For instance, if their planet were not rich in iron, they may have used hydrogen peroxide as a transport method. Hydrogen peroxide is very reactive, and eagerly gives off the extra oxygen molecule to form water. Their 'hemoglobin' could conceivably have evolved into a method that binds an extra oxygen molecule to water in the 'lungs', and then releases it in the tissue.
The very systems that our blood has evolved to protect the functionality of our blood hemoglobin system could be precisely the system that destroys their naturally evolved systems to protect their equivalent to our hemoglobin.
Co-incidentally, this would make their blood just as 'toxic' to humans. It would be like pouring hydrogen peroxide on it. Note, hydrogen peroxide was once used as an antiseptic, until it was found that its oxidizing properties also destroys healthy cell tissue, and delays healing.
Why Does Hydrogen Peroxide Fizz On Cuts?
Despite its negative effect on healthy cells, our bodies' cells
naturally produce hydrogen peroxide when we metabolize food and turn
it into energy. So how can a cell produce something that can destroy
its own walls? That's where catalase steps in: when a cell creates
hydrogen peroxide, it stores it inside the cell's specialized
organelles, called peroxisomes, which contain hydrogen
peroxide-busting catalase. Inside of a peroxisome, hydrogen peroxide
decomposes and is turned into harmless water and oxygen gas .
This quote, incidentally, also posits a credible system for a hydrogen peroxide based blood oxygen delivery system, since the mechanism is already present in some form in earth-based mammals.
Your question did not mention how fast you want the reaction to be.