A pack of 61 large and 97 smaller dogs, based on the animated film The Snowman and the Snowdog, make up a free public art trail.
They will be auctioned after the trail closes on 29 November to raise money for the St Oswald's Hospice charity.
Spokeswoman Jane Hogan said several had been taken to the "vets" with substantial damage.
"The vast majority of the damage is people being too enthusiastic and climbing on them," Ms Hogan said.
"Although they are pretty robust and they are varnished, they will only withstand a certain amount of attention, but we do ask people not to climb on them."

Anthony Jackel, of Port Talbot, also known as 'Betty', is a veteran of Wales campaigns.
The 70-year-old Swansea City fan has bought a train ticket which allows him to follow the team throughout the championships.
He said: "[This way] I can do what I want, I can go where I want".
Of course, the Foreign Office and the Football Supporters' Federation would heavily discourage his travel arrangements - and with good reason.
But he refuses to be put off.
He got the bus to London and the Eurotunnel to Paris before sleeping outside Paris' Gare du Nord station.
Mr Jackel then caught the train to Bordeaux, where I first met him outside the Grande Theatre on Friday. He was instantly conspicuous.
He had the look of a well-weathered man and he wore an equally travelled shirt emblazoned with the Welsh flag over his fleece.
It attracted fans who stopped in the street and shook his hand, complimenting it.
"The last two nights I've slept on the tramlines," he told me. "On one of the big benches. Lovely.
"They've got all the mod cons, a portable toilet where I can go to the toilet, wash my hands and brush my teeth, and I'm up then.
"I have an early morning call: the nightclub shuts at five o'clock [am] and they wake me up."
I asked Mr Jackel why he travelled in a way many people would think was very dangerous.
He said he would rather do it his way than be "herded" around on an organised tourist trip.
"I can do what I want, I can go where I want and I can go home when I want to, I'm not stressed out if I don't get back to my hotel.
"If I'm tired I just go on the first bench or the next place."
He told me he had "followed Swansea all over the world" and had watched Wales since 1962.
"Over the years we have been so near but [also] so far," he said.
The die-hard Swans fan said that, while there are often clashes between rival supporters back home, they had to remain "together" in France.
"We are Welsh out here, we're Welsh, I don't care a monkey's...while we're here we are showing people we can behave."
I asked how excited he was that Wales had made a major finals for the first time in 58 years and he said: "You should have seen my flag, it was too big to carry."
He said he had encouraged younger supporters to come out and support the team.
"I said: 'What if we did win it? What if we did a Leicester [who recently won the English Premier League]? What if we did a Greece [who won Euro 2004]?'"
Mr Jackel pulled a bag on wheels and when I asked him what he had inside I was surprised to find a collection of brand-new Wales soft toys, and trinkets which he said he handed out when people were "nice" to him.
He spoke about football trips to Romania, Spain and numerous other countries.
Mr Jackel said he would follow the team to wherever their tournament came to an end in France.
And while he had been upbeat about his outdoor accommodation, he was less certain about his plans for future tournaments if Wales qualify.
"This might be the last tournament I sleep outside. If I get to Russia [which will host the World Cup in 2018] I'm going to have a bed."
A number of colourful snow dog sculptures have been damaged after being unleashed across Tyne and Wear.
[NEXT_CONCEPT]
While Wales football fans enjoy the hospitality of France for Euro 2016, one will be sleeping rough.