Medals & trophies cannot be recycled however parts may be recyclable.

  1. separate metals from other materials
  2. put metal in recycling bin
  3. you can repurpose ribbon/lanyards for wrapping presents - see Lanyards
  4. chuck anything else you can't repurpose. 

No plastic, glass, or textiles can be recycled from these items.

If you can't separate mixed materials, throw them in the trash.

Can I remove the engraving so it can be reused?

No. 

While you can certainly ask the engraver who sold the item if they offer such a service, so far we have not heard of any company doing this. 

Reuse of such items puts the seller at high risk of passing on a medal/trophy that is  not “shiny and new” which is what buyers are usually looking for when it comes to giving awards of achievement. 

Likewise removing the engraved plaque (typically bonded with strong glues) would damage the object.