So there's only a few ways this could happen 'realistically'.

1) Water comes from underground. This would require a lot of really unlikely scenarios though and probably wouldn't be stable for long. Imagine a tube running from under the oceans all the way to the middle of the mountain ranges. Then having the temperature and pressure force the water to the surface, like real hot springs and geysers (Look to Yellowstone for an example). I say this is unlikely because the immense pressures and extreme distance the water would have to travel would destroy this system.

You could have an extremely large reservoir underground that gets pumped up by geological activity, but it would run out eventually.

2) What Milloupe said. This does already happen, but having it supply a planet's worth of rivers would be unlikely.

3) Special plant life/ trees that pull water vapor out of the air and actually release water into their soil. This would require constant humidity and probably wouldn't work at higher latitudes, if at all.

4) Massive glaciers that are melting over time. Perhaps an ancient lake was lifted by mountains and froze solid. Now it is melting and has carved a path downhill.

The entire concept isn't super plausible without fundamentally changing the physical properties of water though. You'd still have water evaporating and then wanting to condense when the temperature and pressure change.

Would vulcanism work for you?

Vulcanic eruptions can create flowes without rain.

The obvious candidate would be Magma flows.

This is the moon Io, magmaflows of basalt lava crawl over the surface for hundreds of kilometers [6].

If you want liquids other than molten rock, check out cryo-vulcanism [1]. Water, ammonia, methane or some mixed slurries don't exactly make for exiting rivers, but something will undeniably flow downhill.

Finally ocean currents could be considered rivers. While this might not exactly meet your requirements it seemed worth mentioning. Europa would be another moon of Jupiter fitting your conditions in this chase.

While the radial convection currents shown here could be considered to strech the definition of river past its breaking point, the western equatorial flow and the two polar eastwards flows discribed at the end of this article [2] could be seen as rivers. They are compared to our earthly gulf stream.

[1] https://en.m.wikipedia.org/wiki/Cryovolcano

[2] https://www.mpg.de/7655677/Europa-heat-pump-ocean

[6] https://en.m.wikipedia.org/wiki/Volcanology_of_Io

Can't happen with an Earth-like planet

How do you move millions of tons of water from lower elevation to higher elevation? On Earth, the only mechanism to do so is evaporation of water into the atmosphere. If you want no rain or snow, then that is ruled out.

What other mechanisms could possibly move such a large mass of water without evaporating it?

There are some conceivable options, but they all involve the transport of solid or liquid water. In order for either of those to fight against the force of gravity, they would have to be less dense than the fluid medium they are floating in. Therefore, you are left with an "atmosphere" that is denser than either water or ice. This of course, is not really an atmosphere at all, but rather an ocean.

There is no way to get so much mass of liquid or solid water to defy gravity unless you evaporate it. So while you can play with the parameters of what you consider rain or snow, ultimately, it has the transported by evaporation to higher elevations to have a global water cycle.

Tides

(lots of handwavium here)

Imagine that your planet is really very flat, but it has some very wide craters, whose rim is high just a dozen meters above sea level. It also has a massive moon on a very elliptical orbit.

Every N months (Earth months, for that planet it is once a month), when the moon is at the nearest, the tide makes the sea flood into the coast and for dozens kilometers toward the inland, also submerging the craters. After the big tidal wave comes back, a lot of water remains trapped inside the craters, from where it slowly flows again toward the sea (following some paths according to where the rim is lower and the conformation of the land), effectively creating rivers. Such rivers would probably be very wide and slow and also salty.

The fact that the crater floor is usually lower than the surrounding land doesn't matter, since it would contain a lake in this case (whose level is tha same as the surrounding plain). When the tide submerges the crater, the level of the lake would simply rise some tens meters, then the surplus water would flow toward the sea.

Such orography wouldn't last long of course, since the continuous tide would constantly erode the craters, so it is probable that the planet is undergoing a heavy meteorite bombardment, which steadily creates new craters that act as collection basins for the water.

As @AlexP has rightfully pointed out in their comment, "you cannot have a world with liquid water and no rain". Water evaporates, condenses, and precipitates back to the surface.

What you can have, are large areas without rain. On Earth, the area at 30° latitude is comparatively dry. Here lie Southern California, the Sahara, and other areas that have only very little rain. If having your landmasses in that area doesn't suffice and you need a whole world with rivers but without rain, I propose the following:

Terraforming

Your world is in the process of being terraformed. Water is brought down to the planet surface from asteroids. To allow the inhabitants to grow food even while there isn't yet enough water on the planet to allow for rain, the water is brought down from space in high places, from where it flows downhill towards the lowlands, forming rivers. Water from the rivers is diverted through the fields and forests to allow the plants to grow.

Water will evaporate on that world, and eventually it will rain, but the water in the atmosphere won't be enough for there to be rain for as long as you want. All you have to do is slow down the process of bringing water to the planet, and you can have centuries with rivers and without rain.

You'll have to figure out where the breathable atmosphere came from, though.

The first thing which comes to my mind is having a water cycle similar to the one we have on earth, except that instead of raining the water condenses only once near the top of the mountains, and drips directly back to the rivers/glaciers. It's not very different from what we have, and probably happens sometimes on earth, when correct temperature/pressure conditions are met.

A world wrapped in some kind of transparent material like a a solar desalinator

just have the material come down at various locations and water flows will form. It need not be transparent even, you just need some heat source to vaporize the water.

If all the landmasses on your planet were islands, you might have something like a river form between islands that are located close to each other. No precipitation would be necessary, but water would still have a direction and flow (potential for hydro-electricity or mills). I guess if I knew the purpose of rivers in your story I could think of other answers, but other then that I don't think there is an indefinite way to make rivers run in a land of no water fall.

Thanks to some very unique plants :
It's some kind of root or climbing plant, that draw salt water in the oceans and pump it along its veeery big net of planty things and dump it desalinated into lakes or subterranean reservoirs high in altitude.

Why, you ask ? Because they both need the salt (and/or anything else in the ocean) AND some thing that they can only get in the mountains. Sun ? Some other kind of mineral ? Stuff.
And there's a lot of them, everywhere.

Also you could have some kind of bird that lay eggs in the ocean, pump the eggs and salt water in a pocket and then carry all that into the mountains to brood. You'd obviously need a ton of birds, and only get rivers after the mating season.

Besides cryo-volcanism and a water planet mentioned before, thermal vents would also work - geysirs. We'd need to change the way that plate tectonics work to make more water go down when plates overlap. Maybe a material which is more water-rich. And maybe make the plates thinner and with a colder type of lava below - some stones or ores which melt at a few hundred degrees, so the water doesn't evaporate too fast. And also so that the magma cools into stone before it erupts into volcanism, so the water comes out of the volcanoes instead of the magma.

Another already mentioned idea, tides and such from another body, could be made more extrem if the moon was much larger and Earth's crust was predominently of a material which is water rich. The tides could then squeeze together gorges and cracks, which are parts of mountain systems, so water collecting there gets squeezed up and then out.

If artificial means are allowed, make (ancient) wells down to the ground water, and some form of pumps. One way to do the pumping would be to heat up the water with geothermic energy. Underground channels would go from cold to hot places (near magma chambers or such) and then up into slightly higher and warmer lakes, from where it flows through rivers back to the underground reservoirs.

I think a hottish eyeball planet would work for this.

Which is a planet that is tidally locked with it's host star. So the sun points directly over one location all the time. Which obviously makes one side hot and the other cold.

Of course this would have some great effects on the planet's climate. Basically water would evaporate from the hot side and then re-condense on the cool side, the terminator would be the habitable area and liquid would flow from the cold side to the hot side. As the glaciers move from the cold to the warm side they would melt and create rivers.

I don't know if you could do it with no precipitation but "from the ground" it may appear this way. The cold or hot sides would be wastelands devoid of life, so the inhabitants could only venture so far outside of the termination zone. Sun ward would be an sweltering desert with no water, and the shadow side would be a frozen wasteland. So even if it snows in the Dark zone (or whatever you want to call it) no one could live there to see it.

Currently I don't think there is really a consensus on what the climate would really be like, so we have some degree of flexibility here.

Cheers.

This depends of the definition of high places.
The only way this is even slightly plausible is if the whole world is near the freezing point like 33F(possible even lower), land and water. Then evaporation would be small. If all bodies of water were covered, say in ice that would further reduce evaporation. Simply covering a swimming pool locks in 90% of its moisture because it has no where to go.

However, lifeforms like us pesky humans would ruin it because they generate tons of heat. That heat would eventually cause evaporation. Think of a nuclear cooling tower.

If the water was allowed to flow from the ocean into the great lakes, it would eventually travel down the Mississippi and reach the ocean again. Of course all the lakes would probably contain salt water.

From there natural ocean currents would eventually, probably 100's of 1000's of years, carry the same water back to the top and flow again through the great lakes.