The petition on Change.org UK has been signed by more than 79,000 people in 24 hours as thousands claim his environmentalist views are too prominent for BBC neutrality.
Packham has come under fire for campaigning against the killing of birds in the UK considered to be pests, such as crows, woodpigeons and jays. He has also been a vocal supporter of the Extinction Rebellion protests that have caused chaos across London for the last two weeks.
Packham, who hosts shows including Springwatch, Winterwatch and Blue Planet Live, tweeted a photo of two birds hanging from the neck on his gate and said police and lawyers have been informed.
Posting at 4am on Thursday morning, he wrote: “This was my gate this morning (it was vandalised)”
The post was retweeted more than 2,500 times, with comments flooding in to support the presenter.
Packham has long been a campaigner for wildlife but is currently leading Natural England, the Government’s conservation watchdog to revoke a licence permitting the killing of these birds.
The presenter also set up the not-for-profit company, Wild Justice with conservationists Ruth Tingay and Mark Avery to take legal cases on behalf of wildlife against public bodies where they are failing to protect species or habitats.
On its website, Wild Justice says: “Threatened species can not take legal cases in their own names but, with your help, we will stand up for wildlife using the legal system and seeking changes to existing laws.”
Andrew Hayes, who set up the petition on Change.org on Wednesday said: “As an employee of the BBC, Chris Packham should remain impartial and keep his views and beliefs to himself.
“However, he is the face of many anti-hunting campaigns and uses his celeb status as a platform to push his anti-hunting agenda, he has made his goal to ban all kinds of hunting, and country sports and pursuits."
“I feel he is no longer fit to work for the BBC,” he said.
One commenter on the comment page wrote: "This man is abusing his position as a TV presenter to pursue his own beliefs and values.
Another said: “His biased opinions and hatred of countryside management are damaging our country.”
However, fans of the presenter set up a rival petition, which has also hit more than 76,000 signatures.
Rob Niel who set up the petition in support of the presenter said: “Chris Packham is a much loved and respected naturalist and presenter, who campaigns for wildlife and conservation.
“The Countryside Alliance and other organisations are calling for the BBC to sack him, in other words silence him, for speaking out in his campaigns.
“We think his campaigns for wildlife are justified, as they are based on scientific evidence, and he keeps his campaigning and role at the BBC separate.”
One commenter wrote: “Chris has an informed reason for all his views.
“Chris has a right to speak the same as the rest of us plus he is really nice so leave him alone to get on with helping and explaining to the rest of us.
The BBC has been contacted for comment.