I know it’ll be hard to define - just throw out some possibilities. Israel-Palestine goes back to at least 1947. I seem to remember the Koreas have technically been at war since 1950. I’ve been told Ethiopia and Eritrea have similarly been in a ‘state of war’ for decades. India and Pakistan? Israel and Syria? China and Taiwan?
Does the battle of the sexes count? That has been going on forever. 
If you view the China/Taiwan thing as a “current” conflict, then that goes back to 1927, the start of the Chinese Civil War. I’m not sure that there’s been a death due to that conflict since the late '50s, though.
I think we need some definitions as to how to define what you mean by conflict.
As one potential candidate: The IRA is a hundred years old and traces the fight back a hundred years more.
Good vs. Evil.
The wikipedia article on current conflicts offers a couple of ways to measure it. Defining current conflict as one in which there were at least 1000 deaths last year, then it’d be the civil war in Colombia, which dates to 1964.
For ongoing conflicts that have fewer recent casualities, the article cites three that started in 1948: The Korean war, the Israeli-Palestinian thing, and the internal conflict in Burma.
The wikipedia article has a pretty good list that you can adapt to fit your own definitions.
I never even thought to search for this on Wikipedia… Should have known better.
They don’t include China-Taiwan on that list, even though MikeS just said it goes back to 1927. I presume that’s because it has no casualties at all? I’m not sure why the IRA isn’t included either, maybe because the conflict is deemed to be finished? There’s still been sporadic terrorist attacks so you could probably define it as ongoing if you wanted to.
Technically speaking, the Israeli-Arab conflict began in 1920 with the Battle of Tel Hai.
Cats v. dogs
prescriptivist vs descriptivist
vi vs EMACs
Apple vs. IBM?
Coke vs. Pepsi?
Greeks Vs Turks?
They’ve hated each other since oh…B.C.E.
Actually, it dates back to the time of Abraham when Ishmael and Isaac had a falling out with each other.
In the Christian world, the disputes between the Catholic and the Orthodox churches over which is the legitimate One Holy, Catholic, and Apostolic Church have been raging for almost a thousand years (they split in 1054), though they have moved away from violence.
Those were Persians not Turks
USA vs. Latin America, 1823-present.
Volatile India-Pakistan Standoff Enters 11,680th Day. “The threat of nuclear war hangs over the region with no end in sight, just as it has for the past three decades.”
As a point of information, I think you are referring to the Arab/Israeli conflict. According to Wikipedia, the declaration of independence for a state called “Palestine” was in 1988. Before that, again according to Wikipedia, there was an “all Palestine” government which was annulled by a decree of the Egyptian authorities in 1959.
More accurately, it was a rift between their mothers, Sarah and Hagar, a good 15-20 yrs prior.