Avenir is not a standard font or a web font..
Avenir is a system font, which has to be converted to an image during output, to make sure, that it is displayed on every visitor’s machine/device.
We can’t know, why in this special case this font can’t be rasterised. On my machine rasterising of this font works as expected. Perhaps we could find the reason, if you place the text frame onto a newly created site, upload this .muse file to Dropbox or a similar file sharing service, so that wie can have a look.
Simplest method to fix this: Use a standard font or a web font.