ISSUE
Particle system use image texture to influence things like Time, Density, Size, … etc of particles, but any animated texture doesn’t work.
It is a “known” limitation seen on BugTracker even from 2011 … always closed as invalid report. I can understand to this, but from user perspective is current situation very confusing. Even more if for some users it “works” in viewport, but again fail when cached or rendered.
SOLUTION
Since particle system has its own Texture Properties that seems to me is able use only Single image, it would be helpful restricted in some way not working options (movie or image sequence) from User Interface. Or hide all Source row.
Or add some notification …
Thank you for looking to this confusing “thing” that occupy community’s threads for decade.