Why is osmium(VIII) oxide so toxic?
Metallic osmium is harmless but it reacts with oxygen at room temperature, forming volatile osmium tetroxide. Any other osmium compounds are also converted to the tetroxide if oxygen is present. This makes osmium tetroxide the main source of contact with the environment and human body. Exposure to osmium tetroxide cause harm to the skin, eyes, and respiratory system.

So, why makes osmium tetroxide so harmful? (TL;Dr: Because to its oxidization ability)

There is a paper1 which studied the effect of osmium tetroxide in human body and how it can be a lethal bioweapon. The following is the text from the paper:

Osmium tetroxide is highly volatile and penetrates skin readily, and
is very toxic by inhalation, ingestion, and skin contact. Airborne low
concentrations of osmium tetroxide vapor can cause lung congestion and
skin or eye damage.
[...]
Osmium tetroxide is a rapid, indiscriminative oxidizer that does not distinguish between organic tissue and inorganic materials. An
inhalational toxicity study with rabbits proved futile, because of the
rapid reduction of $\ce{OsO4}$ by the skin, hair, mucous membranes,
etc., as well as by the chamber walls. Inhalation, ingestion,
contact with skin and with mucous membranes may all result in severe
consequences. Due to its high vapor pressure (7 mmHg at 20 °C), most
exposures are to vapor. These can cause severe chemical burns to the
eyes, skin and respiratory tract. Very short-term contact with the
vapor may cause lacrimation, accompanied by cough, headaches and
dizziness. $\ce{OsO4}$ may cause irreversible blindness by turning the
cornea black. Symptoms may not be noticeable until several hours after
the exposures. Another severe delayed effect following inhalational
exposure is acute lung injury, which may be followed by
non-cardiogenic pulmonary edema Direct contact with osmium tetroxide
solution will turn the skin black (severe chemical burns due to strong
oxidizing properties). Painful burns or contact dermatitis may
result, depending on the concentration.

The paper compares the LD50 value of osmium tetroxide with other chemical warfare agent and thus proves how lethal it can be as a bioweapon:

The paper also notes that osmium tetroxide can rapidly reduced to relatively inert compounds by e.g. ascorbic acid or polyunsaturated vegetable oils (such as corn oil which contains double/triple bonds. $\ce{OsO4}$ can attack those bonds and can convert to non-toxic vicinal diols).
Reference

Makarovsky, Igor &amp; Markel, Gal &amp; Hoffman, Azik &amp; Schein, Ophir &amp; Finkelstien, Arseny &amp; Brosh-Nissimov, Tal &amp; Tashma, Zeev &amp; Dushnitsky, Tsvika &amp; Eisenkraft, Arik. (2007). Osmium Tetroxide: A New Kind of Weapon. The Israel Medical Association journal : IMAJ. 9. 750-2.
McLaughlin AIG, Milton R, Perry KMA. (1946). Toxic manifestations of osmium tetroxide. Br J Ind Med 3:183-186.