In this video, around the 0:41 mark, the man speaking pronounces petites with the second t lost to a liaison. Is that a regional affectation, or did I just never know this? If this is standard, what is the rule that causes that to be silent?
The man actually pronounces the second t. Even though it is not as clear as the sounds from the beginning of the word, I distinctly hear it. If you play the video at half-speed, you can hear it being merged with the z sound of the s liaison, PETI-tZanimations.
In standard French, the t in that phrase is not silent.
The second /t/ is not lost. It's there, but it merges with the following /s/ (which, itself, is voiced into a /z/ because of the intervocalic context). So in the end you have /pətitzanimasjɔ̃/.
With the e (or es as here) at the end, the t is pronounced, maybe not forcefully, but still there. as already stated, the liaison is more between the s and the following a, so the t sort of gets lost.
I did not watch the video.
It depends. In French, the masculine form of the word is petit (no e at the end). In this case, if the word after "petit" starts with vowel, the "t" would be pronounced (i.e. un petit arbre). If the word after "petit" begins with a consonant, the "t" would be silent (i.e. un petit gateau).
The feminie version (petite) does not have this peculiarity, since there is always a vowel (petite) after it.