The white banyan forest in Zhiben Forest Recreation Area.
The old cypress trees at Shang Baling (you can walk up to them from Fushan near wulai on the 18km Fuba Trail). The old trees at nearby Ma-Kou Ecological Park.
The jungles of Wulai.
The subtropical forests along the Walami Trail in Taitung, one of the best preserved such forests in all Asia.
The bamboo forests around Ruili.
The subtropical forests along the Nanao Old Trail in Nanao.
The mixed forests at Daxueshan.
The mixed forests at Taipingshan.
The pine and hemlock forests along the way to Yushan. Actually there is a belt running down the center of Taiwan above 1800m or so that is pure hemlock and pine: never been cut.
The juniper forests on the back side of Snow Mountain.
The mixed forests along the trail to Nanhudashan. Gorgeous lower forests of maple, oaks, and kiwi, with stretches of huge cypress, pine and hemlock further up.
Fuyuan Forest Recreation Center is supposed to have the best intact camphor forests but I was not impressed.
The tropical forest along the Jinshuiyin National Trail from Fangliao down to dawu in Taitung. One of the best preserved tropical-subtropical areas. Only place I've ever seen a pangolin.
Back side of Yushan on the way to Batonguan has a sublime fir forest where the 30m tall trees grow straight out of the cliffs.
Basically any long hike in the mountains will take you through some gorgeous forest and the higher you get it will be untouched. You'll start in a subtropical mixed forest (evergreens and deciduous trees), move into a cypress, pine and hemlock realm, then higher into fir and finally juniper the last tree species before reaching the alpine region.