In regards to evaporation, it depends on what you consider a river. Because water flowing through a tiny little rill or gully wouldn't necessarily reach a lake. But do you consider that a river? Wadis or ramblas are river beds in arid or semiarid areas, thus, dry. But when it does rain, they normally reach the sea. But if it rains a little bit, then it wouldn't reach.

From Wikipedia's Lost streams of Idaho:

There are two rivers in Idaho named "Lost", the Big Lost River and the Little Lost River. They are often considered separate streams, but both flow into the same depression and become subterranean, feeding the Snake River Aquifer. The rivers are located in Custer County and Butte County, in Idaho in the United States. Via the aquifer and numerous springs, they are tributaries of the Snake River.

Yet another example is Okavango River, as mentioned by Jean-Marie Prival. In the aftermath of the wet season, the Okavango Delta drains into the Thamalakane River, which in turn drains into the Boteti River, which in turn drains into the Ntwetwe Pan. The Ntwetwe Pan is typically a very dry and nearly lifeless salt pan. The Boteti usually dries up before it reaches the Ntwetwe Pan.

There are many other examples of endhoreic basins in the world where the lowest spot is a salt pan that is only occasionally covered with water.

The other category I mentioned are cryptorheic basins. These are typically associated with karst. Karst topography oftentimes results where the surface rock is made of easily dissolved limestone. Much of the drainage in cryptorheic basins is hidden because it occurs underground, hence the crypto prefix.

The upper Danube provides a nice example. It eventually runs dry almost half of the year because its waters sink into the Danube Sinkhole. This sinking happens all year round, but at times the above ground flow remains sufficient to keep the upper Danube flowing. The underground flow eventually merges to form the Radolfzeller Aach, which is a tributary of the Rhine. The underground flow drains to the North Sea while the above ground flow drains to the Black Sea.

There are many other losing streams (streams that lose rather than gain water) in the world. Another example is the Lost River in West Virginia. It disappears under Sandy Ridge and reemerges a couple of miles (three kilometers) away on the other side of the mountain as the headwaters of the Cacapon River. There are several other streams named Lost River in the US.

Eastern Australia also has many disappearing rivers. The Great Artesian Basin is a large cryptorheic basin that ultimately empties to the ocean. Some of the rivers in eastern Australia disappear, not because of evaporation, but because their waters sink to replenish the Great Artesian Basin.

There are many smaller rivers & creeks that do this. For instance, a lot of smaller creeks on the east slope of the Sierra will (seasonally) disappear when they reach the talus slopes at the foot of the mountains.