The British-English phrase to paint the Forth Bridge is used of any never-ending or arduous task.

This phrase alludes to the huge task of maintaining the painted surfaces of the railway bridge over the Firth of Forth, in central Scotland. Designed and built by Benjamin Baker and John Fowler in the late 1880s, this bridge opened on Tuesday 4^th March 1890.

The following, from the West Lothian Courier and Lanarkshire, Stirlingshire, and Mid-Lothian Herald (Bathgate, West Lothian, Scotland) of Friday 13^th September 1901, evoked the task of painting the Forth Bridge:

How the Forth Bridge is Painted.

The Forth Bridge is now in process of receiving its fourth coat of paint since it was erected. Although seemingly a petty detail, this work is really of the first importance, because upon it largely depends the maintenance of the bridge in good condition. As a protection against the corroding influences of the weather, the steel of the immense fabric requires to be kept covered by a coating of special paint composed mainly of oxide and red lead. Ever since the bridge was opened eleven years ago the painting process has gone on continuously. Beginning at the south end, the workmen take three years to cover the entire length of the bridge, and, as three years represent approximately the life of the paint, no sooner are they finished than the men have to begin again.

Before the phrase to paint the Forth Bridge was used of any never-ending or arduous task, the metaphor occurred in the following from The Nonconformist & Independent (London, England) of Thursday 3^rd April 1890:

PROFESSOR STOKES ON SOUL & BODY.
[FROM A CORRESPONDENT.]

There was something of the piquancy attaching to things novel and daring in Professor Stokes’s address at the Finsbury Polytechnic on Sunday afternoon, on “Personal Identity.” […] There can be no doubt that Professor Stokes is right in signalising continuity amid successive changes as that which convinces us that we are the same persons as we always have been. But of course this continuity cannot be realised without memory, and “memory involves thought.” What then, he asks, is thought? That is a question, simple as it looks, on which a quantity of ink has been spent sufficient to paint the Forth Bridge.

The earliest occurrences of the phrase to paint the Forth Bridge that I have found are as follows, in chronological order:

1-: From the column What the Folks are saying, published in The Port-Glasgow Express (Port-Glasgow, Renfrewshire, Scotland) of Wednesday 17^th January 1923:

That there may be several motor buses running in Port-Glasgow in the summer that is on the way.
That it is not likely of course these will be run by Greenock Corporation.
That Port-Glasgow Town Council have opposed the Greenock Provisional Order seeking powers to run omnibuses outwith the western burgh.
That the Tramway Company have been approached with a view to extending the tram lines out to the eastern boundary of the burgh.
That as a first instalment the company will see how much traffic might be forthcoming by running motor buses out to Fyfe Park.
That the company are engaged building a fine big motor bus to be ready for the opening run.
That of course other trips may be arranged when once the Tramway Company scheme gets properly going.
That it may not be generally known that all repairs to the tramway cars are done in the depot at Ladyburn.
That there are commodious workshops at the depot.
That it is surprising to learn that in order to get the fine glossy appearances on the cars no fewer than fifteen coats of paint are necessary.
That this is something like painting the Forth Bridge.
That if an apprentice is started at one side of the bridge by the time he gets to the other he is a jounneyman [sic].

2-: From an article about the work of restoration at Linlithgow Palace, published in the Linlithgowshire Gazette (Linlithgow, West Lothian, Scotland) of Friday 24^th August 1934:

There is a staff of skilled masons constantly employed on the work of restoration, men who have the best interests of the old craft at heart. Like the painting of the Forth Bridge, the restoration work at the Palace continues unceasingly in fair weather or in foul.

3-: From the Sunday Mercury (Birmingham, Warwickshire, England) of Sunday 19^th June 1938:

THESE MEN TIME THE TRAINS
THE INTRICACIES OF TABLE COMPILING
Sunday Mercury Special

TIME-TABLE experts of the main-line railways have just completed their biggest job—the compilation of the Summer time-table, which comes officially into force in the early part of next month.
[…]
“Speeding-up even one single train,” an official told the Sunday Mercury, “may necessitate the re-timing of as many as 250 other trains.
[…]
“Compiling a time-table, in fact, is like painting the Forth Bridge. We’re never really finished. It will not be long now before we’re seriously settling down to work on the winter time-table.”

4-: From London Letter, published in the Courier and Advertiser (Dundee, Angus, Scotland) of Tuesday 23^rd August 1938:

Face-washing of London’s public buildings is again in progress. This job resembles somewhat that of painting the Forth Bridge, because London grime settles so quickly that the end of one operation is never far from the beginning of another.

5-: From the Manchester Evening News (Manchester, Lancashire, England) of Saturday 4^th March 1939:

FOR CHILDREN
The Bridge They Never Stop Painting

HAVE you ever heard people say that something will be finished, “when they stop painting the Forth Bridge?” That means, “never,” because as soon as they finish painting at one end, they start again at the other—which gives some idea of its size!

6-: From the column As R.B.D. Sees It, published in the Illustrated Leicester Chronicle (Leicester, Leicestershire, England) of Saturday 10^th February 1940—the following is about “the impossibility of maintaining the white lines which the blackout necessitated being painted on the roads”:

A small army of men is constantly on the go repainting white lines which never remain effective for more than a week. In many boroughs the job is like that of painting the Forth Bridge: as soon as it is finished it starts all over again.

7-: From In London to-day, published in the Evening Telegraph and Post (Dundee, Angus, Scotland) of Saturday 20^th April 1940:

Painting the White Line.
The task of painting the white line in the middle of the roads in London is like painting the Forth Bridge. As soon as you have finished you have to start all over again.

8-: From an article about the medical examination of school children organised by public authorities, published in the Manchester Evening News (Manchester, Lancashire, England) of Tuesday 14^th May 1940:

Since the war, inspecting school children has been like painting the Forth Bridge. As soon as doctors and nurses had completed their rounds it was time to start again.