You're almost there, but you've got it the wrong way around.
data new;
set input;
if '1jul2015'd <= entry_date <= '30jun2016'd then
output;
run;
Note to keep the date range, you want to output, not delete (note that delete removes observations from the output, not the input!)
data new;
set input;
if '1jul2015'd <= entry_date <= '30jun2016'd;
run;
When you've only got one output dataset, the output action is implied.