tl;dr- No, drinking distilled kerosene doesn't cure cancer as far as any modern research can tell. Some studies have suggested that it might cause cancer in some cases, though the general scientific opinion hasn't found a strong connection between kerosene and cancer under typical conditions.
There's been a lot of research looking for cancer-causing compounds, i.e. carcinogens. Research in this area seems to conclude that kerosene exposure is mostly unconnected to cancer. While I haven't found a reference that explicitly claims that kerosene doesn't cure cancer, presumably this would be noted if such an anomalous result was found.
Pretty much all sources seem to agree that kerosene is toxic and shouldn't be ingested. The primary concern seems to be about kerosene inhalation, which can occur due to aspiration following ingestion. Other sources have noted that kerosene may cause cancer on prolonged skin contact. Finally, it's worth noting that kerosenes are a huge mixture of many different compounds, some of which are believed to cause cancer.
Health safety publications
Regarding carcinogenicity, some studies have pointed toward increased cancer rates in connection to kerosene exposure, though most sources seem to conclude that the potential connection is weak:
An excess of lung cancer was seen in a large cohort of Japanese workers exposed to kerosene, diesel oil, crude petroleum and mineral oil . In another Japanese study, an excess of stomach cancer was observed amongst workers possibly exposed to kerosene, machine oil or grease . Three case-control studies found an association between lung cancer and the use of kerosene stoves for cooking amongst women in Hong Kong; however, no distinction was made between exposure to kerosene per se and exposure to its combustion products . Given that such studies could not attribute the effects to a particular chemical, the IARC evaluated mid-distillate fuel oils as being “not classifiable as to their carcinogenicity to humans (Group 3)” : there is “inadequate evidence” to classify kerosene as a human carcinogen and “limited evidence” for the carcinogenicity of kerosene to experimental animals.
-"Kerosene: Toxicological overview" (2007)
"Compendium of Chemical Hazards: Kerosene (Fuel Oil)", UK Health Protection Agency (HPA) (2006) is an old-but-detailed source of data. Its key points:
Toxicity occurs if kerosene is inhaled while being ingested (aspiration)
Irritating to eyes and skin
Aspiration may cause respiratory irritation
Acute and chronic exposure to kerosene may result in CNS effects including
irritability, restlessness, ataxia, drowsiness, convulsions, coma and death
The most common health effect associated with chronic kerosene exposure is
Kerosene does not have a measurable effect on human reproduction or development
IARC concluded that there was inadequate evidence to classify kerosene as s human carcinogen
Surveying a bunch of sources, I found a few that claimed that kerosene might cause cancer, though most seem to claim that there's no apparent connection with cancer.
Despite the apparent lack of carcinogenicity, kerosene poisoning still has other problems.
"Hydrocarbons Toxicity Treatment & Management" (2015) discusses research on how to treat kerosene ingestion. Most of the concern seems to be related to avoiding aspiration.
Possible origin of cancer-curing stories
If you do enough random testing, you'll find studies all over the place. I found one study that seems to report a ~15% decrease in apparent cancer rates (25 observed malignant neoplasms vs. 29.4 expected) for one group of men who had exposure to kerosene fumes:
3.3 Epidemiological studies and case reports of carcinogenicity to humans
A cohort of 2182 men exposed to jet fuel in the Swedish armed forces was established in 1974 (Seldén & Ahlborg, 1986). Most of the men (86%) were employed in the Air Force, where exposure had been primarily to aviation kerosene, jet fuel, isopropyl nitrate (a starter fuel) and aviation gasoline (for piston engines). Measurements of jet fuel in air at some work places in 1975 and 1976 showed concentrations exceeding 350 mg/m3. Cancer morbidity and mortality in the cohort were followed until 1981 and 1982, respectively, in central registers, with loss to follow-up of less than 0.2%. For Air Force personnel, there was significantly lower total mortality than expected on the basis of national rates (44 observed, 81.1 expected), due to low mortality from cardiovascular diseases. There were 25 malignant neoplasms compared with 29.4 expected. No clear increase in the frequency of cancers at specific sites was seen, even when duration of employment, latency, occupation or type of exposure were taken into consideration. These results were confirmed when the follow-up was continued for a further two years (Seldén & Ahlborg, 1987). (The Working Group noted the short duration of follow-up.)
-"Jet Fuel", p.214-215
Obviously the study referenced above was very weak and far more likely explainable due to other factors, but since it happened a long time ago, people who heard it or studies like it may've jumped to conclusions that didn't pan out.
Reference: What's kerosene?
Kerosene isn't a simple chemical, so samples of kerosene can have significant variation in content. This makes it relatively difficult to discuss compared to chemicals that always have the same composition.
In short, kerosene's like gasoline or diesel. It started out with ancient plant/animal life dying and turning into fossil fuels, which is then pumped out of the ground as crude oil. Crude oil is then split via fractional distillation, where it's boiled in a series of stages, each time at a lower temperature, until it ceases to boil.
So, kerosene's a bunch of decayed organic matter that has a boiling point between 150°C and 275°C. Its exact composition will depend on what the crude oil had in it.
Regardless of crude oil source or processing history, kerosene's major components are branched and straight chain alkanes and naphthenes (cycloalkanes), which normally account for at least 70% by volume. Aromatic hydrocarbons in this boiling range, such as alkylbenzenes (single ring) and alkylnaphthalenes (double ring), do not normally exceed 25% by volume of kerosene streams. Olefins are usually not present at more than 5% by volume.
Kerosene's entry with the CDC describes it as:
Synonyms & Trade Names
Fuel Oil No. 1, Range oil [Note: A refined petroleum solvent (predominantly C9-C16), which typically is 25% normal paraffins, 11% branched paraffins, 30% monocycloparaffins, 12% dicycloparaffins, 1% tricycloparaffins, 16% mononuclear aromatics & 5% dinuclear aromatics.]
-"Kerosene", Centers for Disease Control and Prevention (CDC)
Some components in kerosene are classified as carcinogens
It's worth noting that some of the components in kerosene are classified as carcinogenic. Presumably kerosenes that have large concentrations of carcinogenic compounds would also be carcinogenic, though I'm not aware of any publicly available database that lists detailed breakdowns of kerosenes by the carcinogenicity of each component.