Yhyjyk

On Earth, rivers are possible because rain and snow deposit water in high places. That water then forms rivers when flowing to lower places.

Would it be possible to have the phenomena of rivers flowing into oceans on a planet where raining and snowing do not happen? If so, under what conditions could that happen?

This question is different from this previous one: The Reality of a River World because the accepted answer there proposes a mechanism through which water does not flow from higher to lower places, but only follows tides. I'd like a mechanism to take water to higher places, from where it can flow and form rivers, but not depending on precipitation.

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 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.

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.

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