The Peru trains are operated and advertised as tourist operations, even though locals take them.
The MUNI "Market Street Railway" F line is deliberately a tourist operation.
New Orleans, Lisboa, and Milano are, however, examples of systems which are using antique streetcars in regular service, not as a tourist operation. So I think they win.
The Inclines in Pittsburgh are indeed using original designs (though the cars have been repeatedly repaired/upgraded/rebuilt). This is actually quite common with funiculars (and also aerial trams), because they need custom cars designed specifically for the specific railway; there's no benefit from ordering new if you need completely custom work anyway. As such I'm not sure they count. They usually have no original materials by now. They're "original" only because they've never been thrown out and replaced wholesale. It's like "My grandfather's axe -- it's had five new heads and five new handles, but it's the same axe!". And of course they're not exactly normal railways. I would call them the oldest *designs* of rail-mounted cars in non-tourist use, however.
Now, if we're talking strictly mainline or intercity rail, VIA Rail is arguably running the oldest passenger equipment in the world in non-tourist, non-museum service. But question: is VIA Rail really running intercity rail with those old Budds, or are they actually running a tourist operation? The advertising leads one to wonder.
In short, definitional issues strike immediately -- it really depends on how you define your question.
With regard to "mainline" rail as opposed to streetcars or funiculars, one point to think about is that pretty much every country has disallowed wood-bodied cars on the main lines (with exception sometimes for special tourist or museum operations). This eliminated a lot of the older cars and acts as a sort of "hard cutoff" for the oldest equipment which you'll see in regular intercity service. Wood-bodied cars were removed from urban subways and elevateds by regulation in most places as well. Streetcars often lack this restriction, meaning that the surviving streetcars may be much older.