The phrase to iron out the wrinkles means: to eliminate minor difficulties.

The image is of removing the creases from a piece of fabric, using a hot iron.

Note: Beside to iron out the wrinkles, used in the sense to eliminate minor difficulties, to iron out somebody’s wrinkles was in usage in the sense to smooth somebody, i.e., to make somebody more agreeable. The following, for example, is from Travels in Europe and the East: A Year in England, Scotland, Ireland, Wales, France, Belgium, Holland, Germany, Austria, Italy, Greece, Turkey, Syria, Palestine, and Egypt (New York: Harper & Brothers, 1855), by the U.S. clergyman and travel writer Samuel Irenæus Prime (1812-1885):

The landlord was sole master of the house, and probably he was coarser than he would have been with a wife to iron out his wrinkles.

The earliest figurative uses of the phrase to iron out the wrinkles that I have found are as follows, in chronological order:

1-: From The Model Woman, “the biography of the Perfect Woman”, by Clara Stella, published in the Lancaster Examiner and Herald (Lancaster, Pennsylvania, USA) of Wednesday 31^st December 1851:

Everybody was good with her—to be sure, there were peculiarities in her family, but undoubtedly they all meant well. If people were peevish perhaps they were diseased, or had been roughly used by the world—if maiden ladies had a thousand unnecessary wants, and were continual grumblers, why she had a faculty of ironing out wrinkles, and by uniform kindness, gaining their respect, their confidence, and last of all their love!

2-: From the Sacramento Daily Union (Sacramento, California, USA) of Wednesday 8^th June 1864:

THE CODE OF CONVENIENCE.

A Union victory irons out the wrinkles of the English law. When there is a show of rebel success, the legal obstacles multiply and Minister Adams is compelled to write many notes, and word them with a dainty pen, too, in order to secure a measure of respect for international good faith. The Government is fixed and urgent in demanding evidence which it knows our Minister is unable to furnish. The preparation of iron-clads, pirate craft and other appliances for rebel warfare proceeds without interference, and the official Bulls bellow in a menacing manner about American infractions of commercial rights. The amiable gentry of the “Southern Independence Association” are cheered and comforted by difficulties which they fondly trust may be fanned into war. But let Reuter’s telegraphs flash upon London the news of a crushing victory gained by the Union arms, and, presto! Earl Russell’s policy smooths its wrinkled front. His ways are ways of pleasantness, and all his paths are peace. The Attorney General produces all the legal authority required for the emergency. How the flash of a Union triumph lights up the dark labyrinths of the law!

3[?]-: From The Evening Leader (Wilkes-Barre, Pennsylvania, USA) of Monday 22^nd February 1886—what “the bloody shirt” designates is obscure:

While the Republican caucus Senators are about this thing of worrying into life something to go before the country upon this fall, in the shape of an issue, they might iron out the wrinkles in the bloody shirt. There is precious little left to galvanize outside of that favorite article of partisan commerce.

4-: From The Atlanta Constitution (Atlanta, Georgia, USA) of Saturday 26^th May 1894:

WILLING TO DROP IT.
Endorsements of Mr. Cleveland’s Financial Policy May Be Shelved.
A WAY TO IRON OUT THE WRINKLES
[…]

Washington, May 25.—(Special.)—As a result of the contest in Alabama considerable discussion has been going on among those representatives from the south who are in official position here […].
[…]
[…] It may be definitely stated that from now on no further organized effort will be made to secure expressions of direct approval in the democratic conventions soon to assemble […].
[…]
Since this has been the course agreed upon, it goes without saying that it removes a great many wrinkles from the situation in the south.

5-: From an article about the Republican nomination in Rhode Island 2^nd Congressional District, published in The Evening Bulletin (Providence, Rhode Island, USA) of Saturday 25^th August 1894:

It will certainly be necessary to iron out the wrinkles in the minds of some of the candidates.

6-: From a correspondence from Washington, D.C., dated Saturday 22^nd December 1894, published in The Atlanta Journal (Atlanta, Georgia, USA) of Monday 24^th December 1894:

Mr. Carlisle believes that the more his bill is studied and discussed the better are its chances of passing. His personal popularity with the members of congress, regardless of party and of opinions on the subject of finance, is doing much to iron out the wrinkles that are sure to creep in the way of such an important and far-reaching measure.

7-: From the column Of a Personal Nature, published in The Courier-Journal (Louisville, Kentucky, USA) of Tuesday 11^th June 1895:

Phil T. German, the Assistant Immigrant Inspector at Ellis Island, New York, arrived in the city from New York yesterday, having come home to attend the graduation of his son by the Manual Training High School. He will be in the city two weeks and will be present at the Democratic State convention. Mr. German says that with Democratic harmony the party has good chances. This fact is recognised in New York, he said, and a meeting is soon to be had there by leaders in the Cleveland and Hill-Murphy factions, with a view of ironing out the wrinkles.

8-: From That Depew Boom, published in The Buffalo Enquirer (Buffalo, New York, USA) of Wednesday 12^th June 1895—reprinted from the New York Commercial Advertiser:

Who is better equipped than Chauncey for the Presidency? Where, first of all, shall you find a more courtly, princely gentleman than Chauncey? Where is his peer in oratory, in story telling, in repartee? Where is his equal as a diplomat, a harmonizer, a man who can iron out the wrinkles that now and then come upon the body politic?

9-: From an account of the meeting of the Missouri section of the American Protective Association that was held on Tuesday 17^th March 1896—account published in The Kansas City Times (Kansas City, Missouri, USA) of Wednesday 18^th March 1896:

The judiciary board’s report was called for and read. This was a complete whitewash of President Dearborn. It took up the violent differences which existed between it and the president, and which threatened to rend the order in twain, smoothed them over, ironed out the wrinkles and covered up the ill-smelling mass with profuse verbiage.

10-: From the New-York Tribune (New York City, New York, USA) of Monday 23^rd November 1896:

The Tammany chieftain finds that conditions in politics have changed since his departure. Many appointments for future consultations were made on Saturday night. Dissensions prevail to such an extent that it may require several weeks to iron out the wrinkles on the banner of the Tammany warriors.