The RSPCA centre on Abbey Street has said it is struggling to meet its £5,500-a-week running costs.
Managers have confirmed they will now only take cats and small animals.
Four people will lose their jobs and dogs currently at the shelter will be taken to nearby centres until they can be found new homes.
Centre manager, Malcolm Hall, said they had been running at their 27 dog capacity all year but could not carry on.
"It has been a big decision but at the moment, with the way things are, we just do not have the funding to be able to care for the dogs as we should.
"But the worry is that without the dogs, fewer people will come in to look around and we still need donations to look after all the other animals."
All staff at the shelter have also been asked to reapply for their jobs.