The well-known animal rescue group revealed they have received several calls from people after the Christmas period making the threats.
But DPR say they are inundated with rescue or rehoming requests and are simply trying their best.
A spokesman said “Our volunteers have had several phone calls demanding that we help now or they will SHOOT or let the dog in question go to roam the streets.
“I would like to remind people that these are your pets and although we do offer a Rehoming service it depends on the foster space available and on how quickly we rehome the dogs that are in our care.”
The group said that already 2016 has started the same as every other year in rescue centre across the country.
And they added “We at Donegal Pet Rescue would like to remind people that we are a voluntary organisation who are trying to help as many unwanted pets as we can.
“We appeal to everyone to please have patience. Our volunteers are not here to take abuse or threats from members off the public.”
The group aded that for stray dogs, they do offer a service were they will ADVERTISE for the owner on their Facebook page and contrary to popular believe they are not a collection service for all animals found wandering.
“We would ask people to keep an eye on our page for dogs available for Rehoming and to help share our lost and found album to help reunite pet.
“Please SPAY AND NEUTER your pet and don’t add to the problem.”