You have to walk through 40km of Saudi Arabia to walk from Qatar to UAE.
Brunei is seperated from Indonesia by a strip of Malaysia ; Brunei itself is bisected by some of Malaysia.
The “Wakhan Corridor”, the easternmost part of Afghanistan. The “Corridor” was established in the late 19th century by diplomatic agreements between Britain and Russia, and Afghanistan; in an effort to mitigate the decades-long “cold war” at that time and place, with Russia persistently trying to gain ground and influence in British India, to its south – the narrow strip of Afghanistan was seen as acting as something of a buffer between the two empires. (At the eastern end of the Corridor, Afghanistan also touches on the western extremity of China: the two nations share a very short border, which is at present closed.)
Further African oddities: Angola has a short border with the Republic of the Congo (Congo-Brazzaville) – this is only possible because of the small detached Cabinda enclave of Angola, north of the Congo River’s mouth and abutting on Congo-Brazzaville. The Democratic Republic of the Congo (ex-Zaire / Belgian Congo) has an extremely short seacoast – maybe 10 or 20 km – in between Angola proper, and Angola’s Cabinda enclave.
Not an exceedingly short dividing stretch of territory, but a curiosity which I have always liked: Democratic Republic of Congo, and Mozambique, are not hugely far off touching each other – they are separated by only about 200km of Zambia: which country is basically in two parts, one on either side of this “narrow point”.
Again not a hugely-short divide, but to me intriguing: Poland and Romania come quite near to touching each other – there intervenes, only just over 100km of Ukrainian territory. Prior to World War II, when Poland extended further east and Romania further north, the two countries did share a short border.
Not quite the same, but an almost “touching” (but not trivial such as the use of a river as a border.)… The minimum distance between Papua New Guinea and Australia is 4km … Boigu Island is part of Australia, its 4k over water to a Papua New Guinea land.
Close to home (for those of us in The New World):
Belize is separated from Honduras by a narrow strip of Guatemala.
El Salvador is separated from Nicaragua by a narrow strip of Honduras.
Trivia: if you actually ever manage to cross it, this is the biggest time-zone leap in the world, at three and a half hours. So remember to adjust your watch before you get hauled off by the guards, or shot.
Not quite what was asked, but related - Moldova is separated from the Black Sea coast by a strip of Ukraine about a mile and a quarter wide at its narrowest. And there’s a chunk of Austria that is connected to the rest of Austria by an infinitely small mathematical point.
For that matter Croatia is separated from Croatia by a small strip of Bosnia.
And Oman is separated from Oman by a small strip of the UAE.
How small is small? It looks like Slovakia is about 50 miles wide where it separates Hungary and Poland. Too large to count for this thread?
Are you sure Austria and Switzerland don’t touch at other points?
He said they do touch, that they’re “both big enough to have plenty of contact points either side.” And they do.
Germany and Italy are separated by the pan handle of Austria.
It’s up to the OP as to what counts, but the example given in the OP, the corridor of Afghanistan between Pakistan and Tajikistan, is only about 10 miles wide (17 km).
Personally, I’d be disinclined to count situations like Slovakia and Laos where countries are separated by essentially the entire width of another country, rather than a narrow extension or strip.
A closer example is Italy and Croatia, which are separated by only about 8 miles of Slovenia.
Not as narrow by any means, but I was surprised when I first noticed that Kansas and New Mexico were only 70 miles away from touching at their nearest corners. 
Ah, yes – China’s odd practice of having Beijing time in force for the entire country, even in its far western parts such as Tibet and Xinjiang. Heaven knows how the locals cope with this in practical terms !
Does the DMZ between North and south Korea qualify?
As I remember it, there is about a 3 mile (or KM, can’t remember) strip of land between the two.
Also read somewhere that wildlife in the DMZ is amazing and diverse because few humans get in there.
Don’t know how old my info is… I remember reading this in the early 90’s.
(Off to the web!)
However, the actual border runs down the middle of it. The two Koreas are contiguous.
The Sovereign Military Order of Malta is separated from Italy by only a small piece of the Vatican. But then, the entirety of the Vatican is only a small piece.
New Jersey and Connecticut are also separated by New York. They’re only about ten miles apart without touching.
Another more obscure pair is New York and Rhode Island. Fishers Island, which is part of New York, is only about two miles off the coast of Rhode Island.