Has April 1st ever fallen on Palm Sunday, Good Friday, or Easter Sunday?

Yes, Easter Sunday was on April 1 in the years:

1725, 1804, 1866, 1877, 1923, 1934, 1945, and 1956.

http://www.assa.org.au/edm.html#List19

In these years when Easter was on April 3, Good Friday was April 1: 1763, 1768, 1774, 1825, 1831, 1836, 1904, 1983, 1988, 1994; next time will be 2067.

In these years when Easter was on April 8, Palm Sunday was on April 1: 1793, 1708, 1787, 1792, 1798, 1849, 1855, 1860, 1917, 1928, 2007; next time will be 2012.

http://www.assa.org.au/edm.html

Chart here starts in 1700 - there were obviously many more instances in the preceding centuries!