The phrase to drop the pilot and its variants mean to abandon a trustworthy adviser. The image is of a person who directs the course of a ship.

The Oxford English Dictionary (3^rd edition, 2006) erroneously states that this phrase originated in the following cartoon by the English illustrator and cartoonist John Tenniel (1820-1914), published in Punch, or the London Charivari (London) of 20^th March 1890—this cartoon, titled Dropping the Pilot, was in fact published on 29^th March 1890; it depicts the dismissal of Otto von Bismarck¹ from the Chancellorship of the German Empire by Wilhelm II²:

¹ Otto von Bismarck (1815-98), Prussian minister and German statesman, Chancellor of the German Empire (1871-90)
² Wilhelm II (1859-1941), emperor of Germany (1888-1918)

Although the phrase was certainly popularised by Tenniel’s cartoon (which was frequently mentioned in the 1890s and early 1900s), I have found an earlier instance, as part of an extended metaphor, in The Leeds Mercury (Leeds, Yorkshire) of Thursday 27^th March 1890:

The farewell audience granted to Prince Bismarck yesterday by the Emperor William was the final incident in a crisis which has already become historic. We know nothing as to what passed between the aged counsellor and his Imperial master, who, in the fulness of his youthful confidence, has “dropped his pilot,” and taken over the command of the stately vessel before he has mastered the chart by which he must guide her course; but whatever took place, the act itself constitutes one of the most memorable events in the annals of the Empire.

According to The German Emperor and his Navy, published in The Graphic (London) of Saturday 23^rd April 1892, it was Wilhelm II himself who used a nautical image when dismissing Bismarck from the Chancellorship of the German Empire; this article explains that, “with his heart so given to the sea and its ships”, the head of Wilhelm II is

full of naval imagery, and […] some of the boldest metaphors which have graced his after-dinner speeches […] have been borrowed from the German Ocean and the Baltic […].
And it was again in naval imagery that the Emperor expressed himself when he suddenly dropped his pilot (Bismarck), and wrote:—“The post of officer of watch on the Ship of State has fallen to my lot; but her course remains the same. So now full steam ahead!”

In any case, I have found early uses of the phrase which most probably arose independently from the news of Bismarck’s dismissal.

The earliest is from the proceedings of the General Assembly of the Presbyterian Church in Ireland, published in The Belfast News-Letter (Belfast, Ireland) of Saturday 6^th June 1891—the phrase does not refer to a person but to a policy:

Rev. J. B. Armour (Ballymoney) moved the adoption of the report. He said—On behalf of Dr. Wilson, I beg to move the adoption of the report, with the exception of the postscript, which I hope will be omitted, as we cannot afford to drop the pilot when nearing the rapids (hear, hear)—and in moving the report it is to be understood that the convener is not responsible for anything I may say in urging the Assembly to keep at least to the spirit of a policy which, though discredited and discarded in Ireland where civil liberty is smashed and paganism is crowned, must one day be the retreat of the educational muddle.

The phrase refers to the choice of a religious denomination in the following from Notes from the Outer Hebrides, published in The Daily Free Press (Aberdeen, Scotland) of Thursday 11^th June 1891:

Deputies from the Free, Established, and United Presbyterian Churches, and other evangelistic preachers are sent out to conduct special religious services for the fishermen, which are in most cases, but not always, appropriate to the occasion. It is not to the credit of the Church which they represent nor just to those to whom they minister, to come out to Barra, or any other place, as paid deputies of the Church, with a set of old sermons, no ways fitted to meet the conditions or the wants of the people to whom they minister. Fishermen want something fresh and practical, and with some energy and go in it; and like to see a minister, as did Peter of old, coming to them with his fisher’s coat about him. Such are the deputies wanted for the special services to fishermen. Let the Home Secretaries of the Churches seek such to serve them; otherwise, the people will “drop the pilot,” and prefer a course of their own.