What happens when a mid-ocean ridge sub-ducts underneath continental crust?
We probably can't say that any one thing will happen in every instance.  The particular effects of any one such subduction will probably depend on the event's geometry and the continental rocks doing the subducting.
That said, we can talk about a famous instance of this, when parts of the East Pacific Rise (between the Pacific and Farallon plates) first started going under North America about 30 Ma ago.
Subduction itself is a destructive process: oceanic sediments and water lower the melting point of the crust being subducted, so the subducted plate melts.
So, for the most part, the Pacific-Farallon rifting centers were just swallowed up, and we don't consider the Farallon plate to exist anymore, just its remnants, such as the Juan de Fuca and Cocos plates.  A large transform fault, the San Andreas Fault, formed in the gap.
However, the tendency to rift is still there: A bunch of previously North American continental crust was rifted off onto the underlying Pacific Plate oceanic crust, forming Baja California, and the San Andreas Fault adjusted itself to meet the rift. This was more likely a recent extension of the rift north from the Rivera triple junction than remnant Farallon slab pull under North America.

Source: USGS via Wikimedia Commons - Public Domain