Some bin collections have stopped and rubbish tips are shut in parts of England.
Garden bin and recycling collections are also on hold in many areas, to give household and clinical waste priority.
Some councils reported lower staffing levels due to self-isolation and said employees had been redeployed to support essential services.
Authorities advised people to check council websites and if in doubt put bins out as normal.
Those with coronavirus symptoms should put used tissues in a separate bin bag, tie it securely and leave it for 72 hours before putting it into their wheelie bin, the government has advised.
There were "significant disruptions to services" at Luton Borough Council where 134 members of staff were self-isolating, the Local Democracy Reporting Service said.
"As a result we will be prioritising the collection of black waste bins and green recycling bins," the authority said.
"This means we will be stopping all brown garden waste bins, glass collections and food waste caddies from this week."
Liberal Democrat Barnfield councillor David Franks said the plan seemed "like a recipe for a massive increase in fly-tipping."
In North Yorkshire, all tips have been closed and people urged not to create extra waste by doing DIY.
"While it is tempting to do DIY projects while following the government's advice to stay at home... avoid creating unnecessary waste until tips can reopen," a statement from North Yorkshire County Council said.
Cllr Andrew Lee, executive member for waste management, told residents to hang on to bulky waste until the sites reopen.
"Don't be tempted to fly tip. Fly tipping is an offence and offenders will be prosecuted," he said.
In Hampshire tips have been closed after "record breaking" use over the weekend, despite official warnings about social distancing.
Deputy leader of Hampshire County Council Rob Humby said: "Last weekend over 29,000 people visited Hampshire HWRCs - to put that into context, our busiest Easter bank holiday weekends might see a maximum of 22,000 people."
Residents of Queen's Walk in Stamford, Lincolnshire stuck notes to their bins thanking refuse workers.
Mel Amara-Carnell, 46, posted the idea in her street's WhatApp group.
"If the bins aren't collected the rubbish can build up and end up overflowing - we could then end up with more disease," she said.
"When I first moved to this street we used to meet for drinks on the green and sing carols at Christmas. That gradually faded away.
"These circumstances have brought us all together again."
All Suffolk recycling centres were shut to "avoid non-essential contact and prevent a risk of infection to others", according to LDRS.
People should hold onto bulky items such as furniture, wood and metals as well as garden waste until the sites reopen," Suffolk County Council said.
"Leaving waste outside the centres or anywhere else is classed as fly tipping and is a criminal offence for which people will be prosecuted," it said.
Liverpool City Council suspended all green bin collections.
A spokesman said: "We will prioritise resources to support household waste, recycling and clinical collections."
In Sheffield, Veolia has temporarily stopped green waste collections and warned other bin services may be disrupted as staff self-isolated.
In Leeds, all council tips will be closed, garden waste collections have been suspended and litter bins should not be used as they will not be emptied.
In East Sussex the 10 household recycling sites will close by the end of Monday.
Coventry Council said it had closed London Road tip over social distancing.
Manchester City Council said food and garden bins would be collected fortnightly instead of every week and should be put out alongside general waste.
Bury Council said brown bin collections were cancelled this week and next "while we prioritise emptying grey, green and blue bins instead".
Many councils are asking residents to keep bins on their property and not pavements until collections restart.
Where possible people should compost grass cuttings and hedge trimmings.