The colloquial phrase to tickle the dragon(’s tail) means: to undertake a dangerous or hazardous operation or activity.
—Cf. also the phrase to chase the dragon, which means: to take heroin by heating it and inhaling the fumes.

I have found an isolated early use of to tickle the dragon’s nose in the concluding paragraph of an article about the dysfunctions of Britain’s Foreign Office, published in The Morning Post (London, England) of Tuesday 3^rd December 1867:

How long is this to last? If the Parliament and the country do not see it, a year hence it will be worse than ever. The evil must be resolutely grappled with. Nothing is so silly and dangerous as to tickle a dragon’s nose. When he is once tracked to his cave he should be annihilated.

The phrase to tickle the dragon, and variants, have occasionally been used in relation to:
1-: Chinese parade dragons.
2-: St. George, the patron saint of England, who, according to legend, slew a dragon (feast day: 23^rd April).

1-: Chinese parade dragons—for example in the following from the Marysville Evening Democrat (Marysville, California, USA) of Thursday 10^th March 1910:

Tomorrow the big dragon will be brought forth as it has in years gone by and he will be wiggled through the streets so that all can see […].
[…]
[…] The man that tickles the dragon and urges him on to greater deeds works for nothing and boards himself. He takes grave chances in being devoured and has to be nimble and act a good deal like some of the Marysville prize fighters. Its [sic] an art to tickle the dragon on the nose and get out of the way.

2-: St. George—for example in the following from The Ottawa Evening Journal (Ottawa, Ontario, Canada) of Saturday 23^rd April 1949:

George Drew Tickles English Dragon
SAINT JOHN, NB, April 22.—(CP)—George Drew, National Leader of the Progressive Conservative Party, tonight tickled a dragon in an impromptu speech before the St. George’s Society.
“The English”, said Mr. Drew, “have complained for long they are ruled, governed and controlled by the Scots, Irish and Welsh. We central Canadians are in much the same boat for we are ruled, governed and controlled by Maritimers.”

The phrase to tickle the dragon(’s tail) was used in reference to an apparatus that the Austrian-born British physicist Otto Frisch (1904-1979) designed at the Los Alamos Laboratory, in New Mexico, USA, during the Second World War: The so-called Dragon machine consisted of a slug of highly enriched uranium hydride that was allowed to slide down a piano wire through a cylindrical annulus of highly enriched uranium hydride.

In the address that he gave at the Fast Burst Reactors Conference held at the University of New Mexico, Albuquerque, from Tuesday 28^th to Thursday 30^th January 1969, Otto Frisch explained how the first controlled fission experiment came to be known as the Dragon experiment:
—as transcribed in Experiments with the Dragon Machine (Los Alamos National Laboratory: August 2005), by Richard E. Malenfant:

I made the proposal that we should make an assembly with a hole in the middle, and that the missing portion should then be allowed to drop through the assembly under such conditions that for a few milliseconds the whole assembly would be critical with respect to prompt neutrons. I did a few simple calculations to be sure that this would be feasible, then sent this proposal to the coordinating council. Of course, I was not present when the proposal was discussed, but it was accepted; it was said that Enrico Fermi ¹ nodded his head in a pleased manner and said this was a nice experiment that we ought to try, and I was told that Dick Feynman ², who was present, started to chuckle and to say that this is just like tickling the tail of a sleeping dragon. That is how the experiment was named.

¹ Enrico Fermi (1901-1954) was an Italian-born U.S. physicist.
² Richard Feynman (1918-1988) was a U.S. physicist.

The earliest occurrences that I have found of the phrase to tickle the dragon(’s tail)—used in reference to the apparatus designed by Otto Frisch—are as follows, in chronological order:

1-: From the Buffalo Evening News (Buffalo, New York, USA) of Saturday 27^th August 1949:

‘Tickling Dragon’s Tail’ Proved That the A-Bomb Would Work

Buffalo Evening News New York Bureau.
NEW YORK, Aug. 27.—The story of the “dragon” of Los Alamos, N. M., atomic bomb laboratory—or how a team of scientists put together enough Uranium 235 to find out that an atomic bomb would work and didn’t blow themselves up—was told here Friday.
The story came out in an interview at the Atomic Energy Commission’s regional office with two of the scientists, each still only 25. In effect, it was an experiment which just about triggered an atomic bomb—but didn’t. At Los Alamos, the experiment was termed “tickling the dragon’s tail.”
The test helped confirm a theory that an atomic bomb could be made to explode, instead of just fizzing. And it brought out fundamental knowledge about both the immediate, or “prompt,” release of neutrons when enough U-235 is brought together, and the later, or “delayed,” neutrons which come along from the chain-reaction’s fission products.
All of the work will help develop better atomic piles, the furnaces of the atomic age.
Late in 1944, the test apparatus was enclosed in a chamber with a red dragon painted on its door in an isolated canyon. The apparatus was designed by Professor Otto Frisch, an Austrian-born British citizen who was a pioneer in the discovery of uranium fission. He had come to Los Alamos as a member of a British scientific mission.
In substance, “tickling the dragon’s tail” meant placing one chunk of U-235 on a table, and then dropping enough more to start a chain reaction. In three thousandths of a second the reaction started.
It lasted long enough to let the “prompt” neutrons carry it on. Had it lasted a half second, the material would have exploded. The scientists, hidden behind a five-foot concrete wall, may have survived the explosion but they would have been exposed to a hazardous dose of radiations.
No one could be absolutely sure when the crucial stage—the “prompt” neutron chain reaction—would occur. The tests went on, adding more and more U-235 each time.
Early in 1945 the scientists got so steep a rise of neutrons in the three thousandths of a second that they reached the quadrillion mark with out needing any help from “delayed” neutrons. They had “tickled the dragon’s tail” enough to find that the neutron burst could be boosted far enough to be certain that an atomic bomb would explode instead of fizzing.

2-: From the account, by the International News Service (INS), of the seventh semi-annual report to Congress by the U.S. Atomic Energy Commission, published on Monday 30^th January 1950 in several newspapers—for example in the Springfield Leader and Press (Springfield, Missouri, USA):

ln 1944 scientists risked death to gain vital knowledge by combining two batches of uranium to discover the “critical” point when an atomic explosion would ensue. The venture was called “dragon”—short for the obvious “tickling the dragon’s tail.”

Incidentally, in the Harry Potter series (1997-2007), by the British author J. K. Rowling (born 1965), the motto of Hogwarts School of Witchcraft and Wizardry is Draco Dormiens Nunquam Titillandus, a Latin phrase translating roughly as Never Tickle a Sleeping Dragon.

Heather A. Haas, Professor of Psychology, wrote the following about this Latin phrase in The Wisdom of Wizards—and Muggles and Squibs: Proverb Use in the World of Harry Potter, published in The Journal of American Folklore (University of Illinois Press) of Friday 15^th April 2011:

This phrase may well be related to the traditional English proverb “Let sleeping dogs lie,” but it also bears some resemblance to a proverb introduced in The Hobbit. In that work, the hobbit in question, Bilbo Baggins, says to himself, “Never laugh at live dragons, Bilbo you fool!,” which, Tolkien ³ then writes, “became a favourite saying of his later, and passed into a proverb.”

³ John Ronald Reuel Tolkien (1892-1973) was a British author and philologist, best known for The Hobbit (1937) and the trilogy The Lord of the Rings (1954-55).