As unemployment soars across the country, tenants rights groups and community nonprofits have rallied around an audacious goal: to persuade the government to halt rent and mortgage payments — without back payments accruing — for as long as the economy is battered by the coronavirus. The effort has been brewing on social media, with the hashtag #CancelRent and online video rallies, as well as a smattering of in-person protests, frequently held in cars to maintain social distancing. Representative Alexandria Ocasio-Cortez, a New York Democrat, offered a fervent endorsement of the campaign, encouraging her progressive base to embrace a movement to upend the housing market. “It’s not that it’s impossible to do and it’s not that we can’t do it,” Ms. Ocasio-Cortez said in a live video on her Facebook page on Monday. “We lack enough politicians with political will to actually help people who are tenants and actually help people who are mom-and-pop landlords.” But In New York and other cities, landlords say they too are struggling to pay their bills since many tenants have already been unable to pay rent. They call the advocates’ efforts reckless and say that withholding rent would create cascading consequences, including leaving property owners without the means to pay mortgages and property taxes or to maintain buildings.