The Vesnova orphanage in the central Hlusk district is home to 174 children and young adults with severe disabilities and weak immune systems.
Ten staff members and 13 children were infected as of Thursday.
The orphanage has appealed to the Belarusian government to relocate the sick to hotels and other accommodation.
The Vesnova orphanage is financially supported by Irish charity Chernobyl Children International. It largely cares for abandoned children, and those whose parents are too impoverished to support them.
The charity's founder Adi Roche said the situation there was "critical", and that some of the children were "extremely ill".
Ms Roche said the orphanage had taken measures to try to protect the children, but faced difficulties as they live in "dense" conditions in dormitories.
"There are no painkillers, no cough bottles. We are really worried and we feel we cannot abandon the children," she told Irish Public broadcaster RTE.
The severity of their disabilities means "their chances of survival would be very poor", she added.
Authorities in Belarus confirmed the orphanage outbreak on Friday, but did not disclose the exact number of cases.
There have been 8,773 confirmed coronavirus infections in Belarus, and 67 reported deaths, according to data from Johns Hopkins University.