Does the offline mode on Spotify use internet?

Yes, Spotify's offline mode does use internet but only to a limited extent. When you download songs for offline listening, Spotify actually pre-loads these songs for a short period of time so they can be cached locally. For this, it requires an internet connection, however small it may be.

Once cached, you can then turn off wi-fi and stream them from your device's local drive as many times as you want without using active internet.