Are there any saltwater rivers on Earth?
The water in any river draining the sea is infinitely recycle-able (from rain replenishment), whereas the salt from any terrestrial source is not. So salty rivers, if any, won't exist permanently. Saltwater lakes gain their salinity precisely because they have no outlet, so salt just gets concentrated by evaporation. I don't think there are any truly saline rivers throughout their entire length.

The nearest approximations I can think of are:


Rare ephemeral runoff from emergent salt domes in desert areas.
Freshwater rivers that drain into arid areas where combined evaporation and infiltration gradually reduces the flow to zero. These are more 'very brackish' than truly saline. The Amu Darya in Uzbekistan is one such example. I don't know the salinity of the Jordan as it enters the Dead Sea, but the river is reduced to almost nothing, whilst there are hypersaline springs, and sewage effluent from Amman draining into it.
The only river I can think of that is very highly mineralized from source to sea is only about 3 kilometres long. It is an unnamed river from the volcanic crater on Savo Island, in the Solomon Islands (Southwest Pacific). It is also acidic and boiling hot - quite literally, for most of it's length.