It's not completely impossible, but it requires some unusual conditions.
First: The 'source' of your lava river is always going to be an active volcano, by definition, but volcanos don't always involve tall mountains. There are also Shield Volcanos which form relatively low, broad rises in the terrain where the lava pushes to the surface. It's important to note here that lava doesn't always come from the top of a volcanic mountain either. The heat and pressure come out through the path of least resistance and while the top of a volcanic mountain may be where the lava came out LAST time, subsequent eruptions are as likely as not to come out through cracks and newly opened fissures in the sides of the mountain.
Second: You only get lava when an enormous amount of heat is rising up from the earth's mantle. Generally speaking this occurs in one of two cases: Either you have a 'hot spot' allowing that heat (and magma) to rise up through the middle of a continental plate like the one that's been continuously forming the Hawaiian Islands for the last few million years, OR you have a place where the continental plates are being pushed apart. Now, the biggest example of this is the huge volcanic rift that runs right down the middle of the Atlantic Ocean that's pushing Europe and Africa in one direction and the Americas in the other, but another good example is the Great Rift Valley in Africa that is currently in the process of breaking Somalia off from the rest of Africa and forming two new continental plates.
Both of those examples have historically had occasions of very long-lasting continuous lava flows primarily because instead of going through cycles of cooling and heating where pressure builds up over centuries or millennia to a catastrophic eruption, these volcanoes have a continuous supply of lava constantly rising to surface.
So, all of that basically means the following. To have a river of lava that persists for decades what you need is an area of continuous low-grade volcanic activity that's near the sea, to allow somewhere for all that lava to go instead of building up a big mountain right there. Kilauea isn't venting lava like this right now, but it has done in the past, for decades at a time. You could also get a similar sort of condition from the African rift valley if Saudi Arabia weren't in the way holding the two plates together at the north end.
The main thing is that it's not going to work if you want your river of lava to be out in the middle of a desert somewhere or something. It has to flow into the ocean, and it can't take very long to get there, or the lava will cool and it'll just start forming a big blobby mountain instead.