They added that, while it is tempting to do DIY projects while following the government's advice to stay at home, they urged people to avoid creating unnecessary waste until tips can reopen.
Normal home refuse collections are set to continue.
There were "significant disruptions to services" at Luton Borough Council where 134 members of staff were self-isolating, the Local Democracy Reporting Service said.
The council's bulky waste service is still operating as normal for the removal of large items from your home, as are green waste collections.
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".
Starting immediately the council will pause green garden waste and bulky waste collection services.
"You can of course still take any glass to our public glass recycling sites".
Islanders are already among the best in the country for recycling and the authority is pulling out all the stops to ensure waste collection services continue with minimal disruption at this challenging time. People should hold onto these items until the sites reopen.
Adam Read, external affairs director for Suez, said: 'It is too early for us to see an upward trend in the data for household food waste collections.
Highbridge's recycling centre has temporarily closed and garden waste collections in the Burnham-On-sea area have been suspended due to safety concerns during the Coronavirus outbreak.
"Waste collection crews continue to work on the frontline of this health crisis, potentially putting their own safety at risk to provide this essential and much-valued council service". However, due to staff shortages, the council is warning that it may need to suspend some services at short notice.