Use iD editor for custom tile server.

Editable OSM data is typically stored in an API schema database (most common example provided by the "rails-port" see https://github.com/openstreetmap/openstreetmap-website ) with access provided by the OSM API (see https://wiki.openstreetmap.org/wiki/API_v0.6 ) this is how all modern editing apps interact with the data, including iD.

Tile servers, essentially regardless of the concrete implementation, require that the raw OSM data be converted in to actual geometry objects (in the API schema only the nodes hold location information), and stored in a database capable of handling spatial queries on such objects.

So the answer is a qualified "no", what you can naturally do is replicate the setup of the central OSM system: an instance of the rails-port holding your data and then synchronizing that via the minutely diff mechanism to your tileserver.

This is not that difficult to do, and if we are only talking about a small amount of data doesn't require anything very performant wrt hardware. If you are considering importing the whole planet then it is non-trivial.