Hi Guillaume, there won't be a message as soon as you run into the idle timeout.
This timeout would be controlled by the application pool used for Forms (by default installed as FormsAppPool). This can be increased from IIS Manager > Application Pools > FormsAppPool > Advanced Settings > Idle Time-out; by default the value is 20 minutes.
The reason you wouldn't see a message is because when the application pool hits this idle time-out for the worker process serving the form, it de-spawns the worker process. So when trying to submit the form, there is no process to handle it and a new one is spawned - but this clears the form.
I haven't tried this myself, but you might also try changing the Idle Time-out Action of the application pool to "Suspend" instead of the default "Terminate". You can read more about the Idle Time-out action here. Suspended worker processes "sleep" when hitting the time-out, but if it is common for this to occur, this could take up system resources if there are many worker processes to keep alive.
Edit: It's also possible that these could be caused by the application pool recycling; see this post for more on that. The above was specifically for sessions expiring from idleness.