Do I draw a card in this situation with Oblivion Ring in MTG?
You will not draw a card.

When a spell or ability resolves it resolves fully, then all abilities that were triggered during that resolution are put on the stack immediately after.

Beast Within resolves fully, destroying Oblivion Ring and creating a beast.
Oblivion Ring's 'leaves the battlefield' ability it put on the stack.
The ability resolves, bringing back Wirewood Savage.

Modern versions of Oblivion Ring such as Ixalan&#39;s Binding use a different template.  For those, you would draw a card.

Ixalan's Binding reads


  When Ixalan's Binding enters the battlefield, exile target nonland permanent an opponent controls until Ixalan's Binding leaves the battlefield.  


This creates a one-shot effect (exiling a permanent) that lasts "until" Ixalan's Binding leaves the battlefield.  As per rule 610.3:


  610.3. Some one-shot effects cause an object to change zones “until” a specified event occurs. A second one-shot effect is created immediately after the specified event. This second one-shot effect returns the object to its previous zone.


Wirewood Savage would return to the battlefield immediately after Ixalan's Binding leaves play.  Then you would create a Beast token, and because Wirewood Savage is in play, Wirewood Savage would trigger.