In Christianity monks and nuns practice works of mercy which include caring for the terminally ill or doing what St. Teresa of Calcutta did. What works analogous to what St. Teresa of Calcutta did do Buddhist monks practice?
I believe monks are enjoined to do the same for each other -- Kucchivikara-vatthu: The Monk with Dysentery
But not to act as doctors for lay society in general -- reference
Thus, bhikkhus should not become doctors, full-time nurses or attendants to invalids. They were supposed to live a simple, unencumbered life sustained by sufficient nutriment, appropriate medicines and advanced health standards for the full-time pursuit of spiritual liberation."
"When the Buddha referred to tending the sick, he was referring to fellow monastics. The Commentary,... [has that] a bhikkhu may prescribe and supply medicine to... his parents, to those caring for his parents, to lay-attendants of the monastery and to those residing in the monastery preparing to ordain; a bhikkhu may also [etc.]
Some will act like chaplains occasionally, if invited to -- visiting someone terminally ill and their family in hospital.
Other "acts of mercy" are a fairly rigorous discipline of being harmless, including not killing animals or insects -- and accepting charity (on the theory that it's good for laypeople to practice charity) -- and teaching the doctrine -- and so on, carrying on the Buddhist discipline and relaying the doctrine.
What the Missionaries of Charity did or do is a bit unusual among Christian monastics, though as the French proverb says, "If they didn't exist one would have had to have invented them."
There may be some analous organization among Buddhist monastics, here for example -- Fighting addiction at a Thai monastery -- that's maybe a close parallel but unusual/atypical too though.
My teacher - a Tibetan Buddhist monk - is constantly visiting the sick and dying and consoling the relatives of those who are afflicted among the lay community. I know other monks and nuns who minister in prisons, who counsel death row inmates, who work with struggling youth. You can find monks and nuns who work on social justice issues, who work with interfaith organizations, who are constantly busy teaching and spreading the Dharma to all those who could benefit from it.
It is not hard to find many, many examples of Buddhist monks and nuns both contemporary and throughout history doing such work, but few are as famous as St. Theresa of Calcutta. Generally speaking, Mahayana monks and nuns train there whole lives to be living Bodhisattva's working for the benefit of others.