As of this morning, the Walt Disney World Resort website is no longer accepting dining reservations for June dates, with the exception of some locations in Disney Springs. Also unavailable are tickets for after hours events, dessert parties, and special experiences like Savi's Workshop through the end of June.
With Disney Springs reopening on May 20th, reservations for most dining locations there are continuing to take dining reservations for June dates and beyond.
On Friday, Walt Disney World stopped taking hotel reservations for June dates, with the earliest possible arrival now being July 1st.
Source: Walt Disney World