A “hot button issue” (also “hot-button issue”) is an emotional, explosive, highly divisive political issue (such as abortion). The issue touches a “hot button” (like a button that sets off nuclear weapons). “Hot button issue” has been cited in print since 1970. A highly divisive “hot-button issue” can also be described as a “wedge issue” (since 1976).
   
A central issue (such as a war or an economic depression) is a “paramount issue” (cited in print since 1844). A personal financial issue can be described as a “bread-and-butter issue” (since 1892) or a “pocketbook issue” (since 1900).
 
         
The Free Dictionary
hot button
n. Slang
Something that elicits a strong emotional response or reaction: an issue that became a hot button among younger voters.
 
The Free Dictionary
hot-button issue - an issue that elicits strong emotional reactions
 
Wikipedia: Politicized issue
A politicized issue is a social, economic, theological, spiritual, scientific or legal issue which has become a political issue, as a result of deliberate action or otherwise, whereby people become politically active over that issue.
 
A contemporary example is abortion, an emotive and moral issue which has become a highly contentious legal and political issue in many countries. Terminology relating to such issues often takes the form of loaded language which contrasts with the pejorative terms used in reference to opponents. For example, those who think that abortion should be a legal medical option describe their views as pro-choice, and may label their opponents as “woman haters”. Similarly, those opposed to legalized abortion describe their views as pro-life, and may label their opponents as “baby-killers” or “murderers”.
 
Heavily politicized issues are often called “hot button issues” because almost any position taken is sure to please one group of people and offend another.
 
(Oxford English Dictionary)
hot button n. orig. and chiefly U.S. (a) a strongly emotive, popular, or controversial concern or issue, usually social or political; freq. attrib., esp. in hot-button issue; (b) a desire, need, or concern that motivates people to choose among consumer goods; (also) a product, idea, or form of advertising that exploits such motivation.
1966 N.Y. Times 9 Jan. 7 Dr. Martin E. Marty, Lutheran theologian.., acknowledges that the ‘God Is Dead’ theologians have their finger on the ‘*hot button’.
1966 N.Y. Times 8 Mar. 18 (advt.) The Guts of Salesmanship explains in detail the anatomy of a sales presentation:..how to find your prospects [sic] hot button.
1984 New Yorker 20 Feb. 42/2 Stereo TV, which Jeff Berkowitz..called ‘one of the real hot buttons in terms of consumer purchasing’.
1985 Times Higher Educ. Suppl. 17 Feb. 10/5 Affirmative action or ‘reverse discrimination’, as the Republicans call it, is a ‘hot-button’ issue.
     
Google News Archive
28 February 1970, St. Petersburg (FL) Times, “Jones Assails ‘Opportunists,’” pg. 3B, col. 2:
ADDRESSING the St. Petersburg Woman’s Club, Jones cited cross-busing, drug abuse, hippies and pollution among the “emotional hot button” issues.
 
4 October 1971, New York (NY) Times, “Intellectual Digest Blossoms” by Philip H. Dougherty, pg. 61: 
One is impressed, talking to this young man who speaks of “hot-button issues” and bringing magazines “on stream,” how much highly professional planning C.R.M. (Communication Research Magazine, Inc.—ed.) puts into a new effort.
   
Google Books
Consequences of Failure
By William R. Corson
New York, NY: Norton
1974
Pg. 156:
There will be lower and lower voter turnout in almost all elections except local ones where the “hot button” issue of the moment coincides with election day.
 
29 July 1977, Los Angeles (CA) Times, pt. 4, pg. G5:
Right-wing organizations see this as a ‘hot-button’ issue they can exploit, along with busing and capital punishment and equal rights for women.
 
12 April 1979, Seattle (WA) Daily Times, “New (old) far right is giving liberals the shakes” by Richard W. Larsen, pg. A14, cols. 1-2:
“And now you have being grafted on that structure the ‘hot-button issues,’” she added—the E.R.A., ant-labor, anti-government (“to provide the freedom to business to do things unfettered”), gun control, prayer in school, others.
       
Google News Archive
7 December 1980, Spokane (WA) Spokesman-Review, “Kennedy targeted for defeat” by David Nyhan (Boston Globe, Nov. 28, 1980), pg. G2, col. 1:
“Our surveys show something really remarkable. On a number of hot-button issues, Ted Kennedy could be very, very vulnerable in a Democratic primary—busing, federal funds for abortion, prayer in the schools, a whole list of social issues that really go to the heart of Ted Kennedy’s constituency.”
 
29 August 1982, Los Angeles (CA) Times, “Social Security Issue Heats Up Senate Race” by William Endicott, pt. 1, pg. 1:
By the time Wilson reiterated his ideas and elaborated on them at a Monterey appearance last month, the governor, in deep political trouble after eight years in office and looking for a hot-button issue on which to launch an attack, was ready with commercials accusing the San Diego mayor of “calling for cutbacks” in Social Security.
 
Google Books
Safire’s Political Dictionary
By William Safire
New York, NY: Oxford University Press
2008
Pg. 324:
hot button Word or issue that ignites anger, fear, enthusiasm, or other passionate response.
 
The noun phrase is often hyphenated and used adjectivally in hot-button issue. Such an issue, involving values or morals, lifts an audience out of its seats in protest or approval.
(...)
Since 1981, the collocation has been a staple of political jargon. John L. Stevens, then director of the Republican Governors Association, told The Washington Post in 1981, “There are a whole lot of hot buttons waiting to be pushed; we’re still trying to find out what those buttons are.”
 
WNDU.com
Congressional debate hits on hot button issue: abortion
Rochester, Ind.
Only eight days remain before the 2010 election, and politicians are making sure they’re heard in an effort to gain as many votes as possible come election day.
Posted: 4:46 AM Oct 26, 2010
Reporter: Sarah Rodts