The term "stone age" is not very useful, since different human societies around the globe have used stone for tools at different times. It's especially problematic because during the so-called "stone age," many human societies developed pottery, which can be used to indirectly heat liquids.
The question of whether or not people drank "hot water" is not something that can be ascertained from archaeological records. Human remains don't bear the evidence of the temperature of the liquids we drink regularly, and even-- hypothetically-- if you ended someone's life by pouring boiling water down their throat, their skeletal remains would not indicate that as the cause of death because the hot water would not visibly affect the bones.
That said... interestingly, one of the most ubiquitous materials on human occupation sites is what archaeologists call "fire cracked rock," which is basically what you get when you use hot stones as a heat transfer method. Repeated use of rocks to indirectly heat causes cracking from expanding and shrinking, and when liquid gets into those cracks, additional heating causes the water to expand within the crack. The results can be anything from just falling apart to a small but fairly spectacular explosion. Fire-cracked rock is recognizable because the cracks typically are different from the sorts of cracks created from direct impact, and the stone itself tends to take on a reddish tint resulting from the heat-related oxidation of iron in the stone.
The evidence of widespread use of indirect stone heating (including boiling / cooking) suggests that humans used indirect heating for a very long time. It may not only have been used for cooking-- a hot rock can emanate heat for a while, and could be used to heat a small enclosed space-- but it most likely was used for some degree of cooking for a long time.
Today, boiling drinking water is practiced in many less developed regions (or in developed areas following some kinds of spills) in order to kill parasites and destroy other water-borne pathogens. We certainly recognize from a variety of archaeological / bioarchaeological sources that most people in our more ancient past suffered from various degrees of parasitic infestation. It's unclear to what extent ancient people may have understood that boiling water can reduce the potential to contract water-borne diseases. I suspect that, like many things, we don't really give ancient people enough credit for recognizing the positive effects of such acts, even if they didn't understand the mechanism. But we don't really have any strong (or weak) evidence of drinking water being boiled.
So, the next question: how is the water held for indirect stone boiling? Pottery was invented at different times in different parts of the world, from prior to the end of the last glacial maximum to only a few thousand years ago. Early pottery generally doesn't look to have been all that durable, and probably wasn't suited for placing on a fire, so was probably used for stone boiling as discussed above. But pottery is not the only way to hold water.
Water can also be contained in...
a hole dug in the ground or pounded into exposed bedrock (possibly lined with an animal hide, but not necessarily)
vessels made from perishable materials like a dried gourd or bark, or a burned out piece of wood
tightly woven baskets, with the interior coated with pitch for extra waterproofing
durable vessels can be ground out of soft stone (e.g., soapstone / steatite)
Any of these, and potentially several of them, could be used as a means of holding water for stone boiling. Given the apparent antiquity of indirect stone heating based on the ages of archaeological sites from whence fire cracked rock has been recovered, it is likely that all of these were used at some point around the world for heating water.
Whether it was drunk hot is simply not something we can work out from the archaeological record, however.
Tales of drinking tea in Canada date back to Jacques Cartier's second voyage to North America. In 1535 he wintered on the St Lawrence River and when he and his men started getting sick from scurvy the Huron taught Cartier and his men to boil spruce needles and drink the tea to cure themselves.
Further west were tribes who's names were derived from using stones to cook their food and to boil soups and teas. The name Assiniboine literally translates to Stone Cree and until recently the natives living just east of the Rockies were collectively called Stoney Indians. West of the Rockies the Beaver and Sekahni Indians drank a soup made from the fleshy inner bark stripped from pine and spruce trees during times of starvation during long winter months and every tribe between the great lakes and Coastal Mountains has tales of boiling their moccasins or leather clothes to make a soup during hungry times during the long winter months.
In Northern and North Central British Columbia the natives used teas for a variety of medicinal purposes. . The stalks of devil's club was boiled and drank to alleviate the aches from arthritis. Bark from poplar trees was boiled into a tea and drank cold for a deworming medicine. Willow leaves were steeped in hot water which released a natural form of aspirin to releave general aches and pains. The flower of the yarrow plant was boiled and then the flower was bound over an infected wound to draw out the poison. This was also used with some success as a method of curing blood poisoning. The natives living in the far Northern regions of British Columbia and Southern Yukon Territory boiled a mixture of Cariboo Grass and Balsam Bark and drank this to cure cancer. Natives living in Northern Ontario also made a tea of various roots and leaves which they used to cure cancer. This concoction contained some plants that were brought to North America by European immigrants so this implies that the Indians were trying new Plants and making new recipes as time progressed. For non medicinal purposes as well as to prevent scurvy they made teas from spruce needles and also from Rose Hips. Blueberry leaves were drank as a tea for no other reason but because it tasted well. The Beaver Indians also dug up a root which they boiled, added sugar to then let ferment into a weak alcoholic beverage. I remember drinking some that was made by my dads employees back when I was a child and thinking it was a putrid concoction. They showed me what plant it was but as I was only about 11 years old I wasn't paying enough attention to remember what they showed me.
There are many other plants that the North American natives used for making teas from, both for medicinal and recreational purposes. The types and uses varied from region to region. There is a fungus called Chaga that grows on birch and poplar trees. The one that grows on birch has been receiving much attention today with a tea made from it being claimed to be a cure all for diabetes, high blood pressure, psoriasis and a number of other ailments. This tea has been investigated by some labs which find there is some merit to these claims. The use of Chaga as a medicine originated from Siberia and has no history here in North America that I have been able to find. Though it was not used as a medicine here the fungus was commonly used by the natives here. The fungus has a hard Woody crust and a soft pulpy inside. The soft internals of the chaga is moist and slowly smoulders so the Beaver and Sekahni Indians would use it as a method of carrying fire with them on their travels. A large piece of chaga can smoulder for up to two days and when they wanted to light a fire they would just hold some tinder against it and blow on it until the tinder caught on fire.
Each tribe took advantage of what materials were most abundant to them when making their cooking vessels. The plains Indians used buffalo hides. Those from Northern British Columbia made water proof baskets from spruce bark ( I have a friend who is in her early 90's who still makes these). Natives from central and coastal Southern British Columbia wove water tight baskets from cedar bark. The Chinook and Coastal Indians built boxes from split cedar which they used to cook soups in. The water in almost all of these pouches, baskets and boxes was heated by placing rocks heated in a fire inside them. They could also boil water in leather pouches by placing them directly over the fire.... but the pouch had to be soaked first and the fire could not reach higher than the level of the water or the hide would burn.
This works on the same principle as the old bar trick where you tell somebody you would give them a twenty, fifty or hundred dollar bill if they can burn a hole through it with a lit cigarette while holding it tight to their arm. It is almost impossible to do because the bill will not burn until the skin beneath it has been scorched burnt away from the arm. I say almost impossible to do because I had a roommate back in college who was in a bad accident and lost all feeling in his left arm. When he was broke and wanted to go on a bender he accepted this challenge for a hundred dollar bill and won them another person didn't believe it and made the same bet and lost as well. Beer was was a dollar a glass back then and we had a wild night on the town with his winnings. The next morning he had two holes in his forearm that were a half inch deep and almost an inch wide.
Stories of using stones for cooking were written by John Jewitt who wrote of his time spent as Macquinna's slave in Friendly Cove during the early 1800's, Father Lafiteau, who wrote "The Customs of the American Indian" back in the 1600's, Henry Kelsey's Journal which he wrote during his trip into the Canadian Prairies in 1690, Alexander Henry the Elder's journals in which he recorded his travels as a fur trader in the Canadian Prairies during the late 1700's and Father A. C. Morice's notes written while he lived among the Carrier Indians during the latter years of the 1800's.