Is it possible to post video on Facebook user timeline using Facebook Autopost module?
No. Not without building upon it. While the version of the FB PHP SDK (3.2.3) that the module relies on supports uploading video, there is no support for it in the module. 

Looking at the current module, it looks possible to build in support. For example, take the scenario where the module supports uploading an image to the user timeline:


The rule action triggers rules_action_publish_to_facebook_timeline() which retrieves (in facebook_autopost_entity()) a FBAutopostEntityPhoto object which is extended from the FBAutoPostEntity class.
When FBAutoPostEntity::publishEntity() is called to publish the entity, it calls a parent method FBAutopost::publish() which checks a list of types that does include a video type. 


Presumably, one could extend another class (e.g. FBAutopostEntityVideo) from FBAutoPostEntity that would add the various details for base classes that would allow it to plug in to the underlying SDK.

To sum it up, the existing module doesn't support however there's some support at lower levels in the code. That comes with the caveat that there might be more overhead to add (e.g. is there more to defining a Video entity than just extending the FBAutoPostEntity class) than what I mentioned here.