Calls for internships and fellowships are announced via www.incca.org in the News section or under the menu item Opportunities.
Below are links to organisations who host interns and fellows on a regular basis:
Tate
The Getty Conservation Institute, Los Angeles, USA
The Metropolitan Museum of Art, New York, USA
The National Gallery of Art, Washington, USA
Smithsonian Museum Conservation Institute (MCI), Suitland, USA