"The health, safety and wellbeing of our Council employees and service users is paramount".
Community recycling centres are also closed.
Carron McDiarmid, a council executive chief officer, said: "While we are doing our best to maintain our regular waste and recycling collections we need to be realistic and plan for staff shortages". To help avoid this, Dorset residents should recycle correctly, minimise the waste they produce and observe health guidance if they have symptoms of COVID-19.More news: First known child death in USA from coronavirus
Somerset Waste Partnership, which manages household waste services in the county, has closed all of Somerset's 16 recycling centres.
The council said: "Personal waste (such as used tissues) and disposable cleaning cloths should be stored securely within disposable rubbish bags.These bags should be placed into another bag, tied securely and kept separate from other waste.This should be put aside for at least 72 hours before being put out for collection on your next collection day".
Please do not leave any waste outside the recycling centres - leaving waste outside the recycling centres or anywhere else is classed as fly tipping and is a criminal offence for which people will be prosecuted.More news: Why Oprah and Stedman quarantined separately
Try to minimise the amount of waste your household produces.
The sites are operated by waste contractor Suez which will still be operating the waste treatment sites where council vehicles deliver waste and recycling from household bin collections.
Starting immediately the council will pause green garden waste and bulky waste collection services.More news: Jamal Murray's girlfriend asks for help after alleged Instagram sex-tape leak
Bury Council said brown bin collections were cancelled this week and next "while we prioritise emptying grey, green and blue bins instead".