If implemented, Section 40 of the Crime and Courts Act would mean any title that is not part of a Royal Charter-backed press regulator would have to pay both sides' costs if they are taken to court – even if the claimants' allegations are proved to be unfounded.
It is supported by former F1 boss Max Moseley, who funds new press regulator Impress under a Royal Charter.
The legislation could destroy the free press and lead to hundreds of local newspapers going out of business.
A Government consultation on Section 40 is set to end on January 10, after which culture secretary Karen Bradley will decide whether to press ahead with the law.
Martin Wright, editor of the Shropshire Star, said: "We need as many readers as possible to respond to the Government's consultation on these grossly unfair proposals to ensure that regional papers like the Shropshire Star can continue to hold those in power to account.
"The implementation of Section 40 would represent a very real threat to democracy – and we need readers' support to ensure the Government understands the strength of feeling on this issue."
Politicians from both sides of the political divide have criticised Section 40, including Government Chief Whip Gavin Williamson.
South Staffordshire MP Mr Williamson said: "Section 40 will quite simply destroy and close so many local newspapers and actually, it doesn't do what is required in terms of better regulation of the press.
"Keeping the local newspaper industry going is vital to our democratic process and is something we should all fight for."
Most newspapers, including the Shropshire Star, are regulated by IPSO, which has said it will not apply for a Royal Charter.
To see the Government's consultation and to respond to it, visit gov.uk/government/consultations/consultation-on-the-leveson-inquiry-and-its-implementation