Benzodioxole is just the IUPAC nomenclature for methylenedioxybenzene (benzodioxole itself is just that, a phenyl ring with a methylenedioxy bridge forming another, non-aromatic ring attached to the benzene ring. Nothing whatsoever to do with dioxin, other than a slight phonetic similarity. And as for dioxin, it is only certain dioxins that are toxic, dioxin itself has NO toxicity in common with polychlorinated dibenzo-1,4-dioxins, its just 1,4-dioxocyclohexa-2,6-diene (the ether solvent dioxane with two double bonds in the ring, one between the #2 and #3 carbons and the other between #5 and #6 positions. Certain substituted dioxins are toxic, and nasty as all hell but dioxin itself is not particularly toxic.
The (extremely) nasty shit known commonly as 'dioxin' is 2,3,7,8-tetrachlorodibenzo-1,4-dioxin (p-dioxin is a synonym for the 1,4-dioxin ring system, where the two oxygens are para to each other, the other isomer being 1,2-dioxin, with the oxygens ortho to each other, this is equivalent to an organoperoxide and its extremely unstable, as are many organic peroxides, a lot of them are explosive, such as the acetone peroxides and ascaridole, found as a natural component of the essential oil of the wormseed, or epazote plant, and not only one of the only organoperoxides in the natural world, but probably the only plant-derived high explosive (its a sensitive and unstable primary explosive in the pure form. Plenty plants have exploding seedpods, but this is a physical process, explosive dehiscence, and does not involve an actual chemically-powered explosion, its purely mechanical, and not a true detonation or deflagration, Common broom is a good example of this (albeit very tame compared to some plants, such as the tropical Hura tree, the large seed capsules of this apparently go off with such a bang that unfamiliar people near the tree when the seed capsules go off think they are being shot at and dive for cover

), with broom the seed pods dry out unevenly, causing tension and when the process is sufficiently advanced the uneven torsion between the two sides of the pod, due to drying of the sides at an uneven rate caused by sun exposure affecting the side facing the sun to dry faster than the side in shade, it splits, catapulting the seeds with an explosive burst of movement, so as to ensure they are shot off away from the parent plant so as to allow the broom plants a chance at getting enough light to survive and allow for growth rather than be suffocated by the shade of the parent)
Whilst dioxin itself (1,4 dioxin, p-dioxin) is devoid of the toxicity found in TCDD, the commonly referred to as 'dioxin' or some other substituted dibenzodioxins, one probably shouldn't drink it, otherwise, unless it peroxidizes (its an ether, many ethers form organoperoxides and organic hydroperoxides that are extremely shock, heat, friction and slightly impolite speech-sensitive explosives of the primary and generally speaking, utterly psychotic, bastardly and bad-tempered kind, that will go off at the slightest provocation. NOT a pretty sight if it happens to be in a large bottle of ether, THF etc. and like they are prone to doing, vaporising and forming peroxides in the screw-thread of the bottlecap. Thereby forming a bomb, that will almost certainly go off if someone attempts to open the bottle of ether/THF/dioxane or whatever due to the friction occuring when opening the now booby-trapped ether container. As a result, most ether is shipped from the supplier to the end user with an inhibitor such as BHT in it in small quantities, although where this would interfere with the end use of the ether/THF or other etherial solvent, copper wire can be used, and the ether should be tested regularly for peroxides using paper test strips impregnated with starch and some potassium iodide as the oxidizing nature of the peroxides oxidizes iodide to elemental iodine that reacts with the starch to give the same indigo-bluish black color one gets after dripping tincture of iodine on a potato.
These peroxides are the reason one shouldn't ever let ethers boil away to dryness (it will likely set off any peroxide formed, making a ruined, shattered mess of your labware and quite possibly, a ruined mess of your face as well

) or boil down to a small amount, because it concentrates any peroxides present in a lesser concentration down into that small volume, resulting in a much more concentrated solution which is then much more likely to again, remove your fingers/eyes/face/head/arms and legs, depending on the quantity of peroxidized ether being worked with, or if it goes off anyway in a large volume of solvent, there is of course, the potential for a huge fireball in addition to the explosion)
The solution is again, testing a sample with starch/iodide paper which can be made easily using potato starch or clothes starch, which conveniently can be obtained as a spray, making it easy to coat/soak the paper, which can first be impregnated with a solution of sodium or potassium iodide and then dipped in the liquor from boiled potatoes, boiled down to a more concentrated solution, and used for testing. Storage of ether with a few twists of copper wire helps prevent peroxidation by destroying them as they form.