This is a notoriously tricky item to dispose of. If it works, please pass it along for reuse at a community thrift, NextDoor, Facebook Buy Nothing Group in your city, or such.
Consider repair before disposal. The internet has many DIY video tutorials on fixing for the common reasons these stop working. Your oil-filled heater brand's website will have repair how-to information and can be contacted regarding issues (ex. DeLonghi). Alternately, contact a local heater repair company regarding repair or refurbishing.
Oil-filled heaters are considered a major appliance under the Public Resources Code. Click here for DTSC's list of Certified Appliance Recyclers. Appliance recyclers have contractors to remove the mineral oil. Most bulky and curbside programs accept appliances. This metal appliance is not accepted by Household Hazardous Waste facilities, however, the drained mineral oil from the heaters is accepted in a resealable, labeled bottle. Contact a metal recycler, appliance recycler, and/or your bulky pick-up service provider for recycling of broken oil-filled heaters.