Does silence affect hero passives?

Silence will not disable passives or items. Only prevent people from actively casting a spell or interrupt a spell they are currently channelling. EDIT: Silence will disable any passive invisibility though (Rikimaru's Ultimate, Moonlight Shadow or Broodmother in her web)

There are 2 skills in the game that WILL disable passives though. Doom's doom, and Legion Commander's duel. These are considered "mutes" and disable EVERYTHING, including items (both passives granted by the item and active item abilities). Although, Legion Commander's doesn't disable every passive in the game. So Doom is still the strongest in that regard.

As if it wasn't confusing enough another variation is Disrupter's ultimate. With the Aghs upgrade it'll disable spell casting AND item usage but NOT passives.

Here is a breakdown of it including all the inclusions and exclusions between LC and Doom's mute: http://dota2.gamepedia.com/Silence

Only passives invisibilities are disabled by silencing (This include Rikimaru's ultimate, Broodmother in her web and the Invisibility granted by Mirana's Moonlight Shadow).

The only way to "silence" a passive is to Mute (with Doom or Duel). You can have a full list of the passives abilities disabled by Mute on the link