The Fire Service found a well-developed blaze at the Welcome Organisation's building in St Peter's Square North in the west of the city at around 3.30am on Sunday.
It is understood the building was broken into and the fire started at the back of the property.
Sandra Moore, who runs the centre, said the damage has hit some of Belfast's most vulnerable people.
"We will not be able to use the centre again such is the level of damage that has been done," she told UTV.
"Half of the building has suffered and the roof is gone in one half of the building. It's literally gutted.
"It's a shell and the other end of it is very badly smoke and scorch damaged."
She said they are currently working to ensure the people who use the building have access to showers and heat and are safe.
But Ms Moore said the centre is more than just a facility to those who need help.
"This is their place in the community," she said.
"This is probably the closest a lot of people have to a family and you can guess what it would be like to lose your home or somewhere that you familiar with."
Police are investigating the cause of the blaze and have appealed for witnesses.
Meanwhile firefighters tackled a blaze at a flat in Belfast city centre.
It happened in the fifth floor of the apartment building at College Square North at around 5.45am on Sunday.
Six fire engines were tasked to the scene after reports of large quantities of smoke. The fire was quickly brought under control.
No-one was at home, although a number of other residents were evacuated for a time.
Investigations into the cause of the blaze are continuing.