<p>What materials work well for lubricating moving PLA, ABS, or PETG parts? I'm talking items like the
the <a href="https://www.thingiverse.com/thing:53451" rel="nofollow noreferrer">Gear Bearings</a> or <a href="https://www.thingiverse.com/thing:4575774" rel="nofollow noreferrer">Print in Place Engine</a>.</p>
<p>I've played with a few lubricants on my own, including hand lotion, trumpet valve oil, and carmex/vaseline. Of these, the vaseline has worked best for me so far, but I'd like to hear what has worked well for others, or especially if there's anyone here who understands the chemistry involved and could explain what to look for in different situations.</p>

When dealing with lubrication of plastics, any solvent or reactive substance is to be avoided. Petroleum is risky and Vaseline™ is a brand name for petroleum jelly.
I've had quite good results using inert lubrication such as PTFE and silicone based lubes. PTFE is the generic term for
Teflon­™
and is quite a good lubricant. There are both silicone and PTFE greases for higher viscosity applications.
From the Teflon™ link:
Teflon's amazing properties are down to its structure. Like most
polymers, Teflon has a carbon-based chain. However, instead of
reactive C-H bonds which occur in most polymers, Teflon has all its
hydrogens replaced by fluorines. These strong C-F bonds are extremely
resistant to attack by any other reagents, making Teflon very inert.
This means that no other molecules will react with or stick to Teflon.
The exception is Teflon itself, which will stick to itself quite
readily, forming thick layers or solid blocks. With a friction
coefficient of <0.1, Teflon has the second lowest friction coefficient
(surpassed only by diamond-like carbon), which makes it perfect for
non-stick items e.g. pans. DuPont invented the non-stick pan coated
with Teflon in 1956 and have manufactured it ever since. Teflon
coatings are so slippery that they are the only material that a gecko
cannot stick to.
Who knew that gecko testing was a thing?
Wikipedia
for silicone grease:
Although silicones are normally assumed to be chemically inert,
several historically significant compounds have resulted from
unintended reactions with silicones.
Powdered graphite is also a good lubricant if one can tolerate loose powder in some constructions.
I've read of others using lithium grease, but not for plastic lubrication.