Is my understanding correct that all involved parties must be online?

The reference server is an Android app so yes, that is probably the point of the default design, but reading the README I believe you can also use a more traditional server-to-server setup:

    DeltaChat/ArcaneChat Integration
    
    DeltaChat and ArcaneChat are perfect companions for Tyr. These are messengers that use email protocols but provide modern chat interfaces. When you configure DeltaChat/ArcaneChat to use Tyr's local server:
    
        1. DeltaChat/ArcaneChat sends messages via SMTP to Tyr
        2. Tyr wraps them in Yggmail protocol and sends through Yggdrasil
        3. The recipient's Tyr receives the message via Yggdrasil
        4. Their DeltaChat/ArcaneChat fetches it via IMAP from their local Tyr
        5. All this happens peer-to-peer, with no central servers

Such a setup could be useful if you find people around you using Tyr and you're losing messages because your phone kills the app, though a PoC like this probably won't have much of a network effect.