Can a wooden indoor rocking chair be put outside?

The caning will eventually deteriorate even if you've used polyurethane on the chair.

The seat will ruin . Sometimes depending upon weave and actual material if try poly coat it It will get brittle and crack.

Not a good idea to put it outside. The cane will sag and the chair will rot.

Agree with Barbara. If the chair was not originally intended for outside use, it will soon get wrecked due to the humidity, moisture, and temperature changes.

You could use Marine Spar Varnish, but all the joints will gradually come apart if exposed to moisture, whether from rain or humidity. Hot and cold fluctuations will also ruin the joints or any seams.

I wouldn't put a caned chair outside unless its on the porch, cause it wasn't made for exterior use.