"What kind of systems do we need to coarse-grain to observe interesting phenomena?"
Systems that would otherwise be too large and/or complicated to simulate in a reasonable amount of time and/or with a reasonable amount of computational resources.
Basically if you want to see $t$ seconds of dynamics for $N$ atoms, and your computational-resources/time is only enough to do $M$ atoms where $M \lll N$, then you could groups many atoms together into what are called "pseudo-atoms" and then you can do the simulation on the $M$ resulting pseudo-atoms (with a slight or not-so-slight loss of accuracy). For example when simulating a long DNA molecule, you might group each nucleotide as a single pseudo-atom, which seems quite reasonable since each nucleotide on its own will likely stick together as a unit and it's only the nucleotide's relative position with respect to other nucleotides that might change, the the DNA molecule comes into contact with DNA helicase, for example. This grouping of multiple atoms into a pseudo-atom, is all coarse-graining is.
"When I go to talks, I hear the statement "coarse-graining is required to access spatio-temporal scales to observe important dynamical behavior for stimuli-responsive systems" before they go on about coarse-graining methods and so on."
I guess for those "stimuli-responsive systems" the combination of the spatial scale ($N$) and temporal scale ($t$) is large enough that coarse-graining is necessary to observe whatever those people think is "important" behavior.
"My question is, what are these stimuli-responsive systems?"
That term is independent of the term coarse-graining. You can coarse-grain any system, whether or not it's stimuli-responsive, and you can simulate a stimuli-responsive system with regular non-coarse-grained MD if you have enough time and computational resources.
If you're interested in what stimuli-responsive systems are, perhaps you could ask a question specifically about that (since the title of this question is, "what kind of systems require coarse-graining"). The review paper published only 4 years ago, with title "Stimuli-responsive systems and their applications" may be a good start!
The Wikipedia article on Coarse-graining never even once uses the word stimuli or stimulus or response or responsive.
"I have run simulations of PNIPAM and have seen its thermoresponsive behavior (coil-to-globule) collapse around 305 K, but all it took was a 100 ns simulation for about 35k particles, tops."
It sounds like you don't need to be coarse-graining at all to see that behavior in that system! 35k particles is still well within the limits of what we can do with non-coarse-grained MD!
"I assume we need to perform CG on more complicated systems, but specific to stimuli-response, what are these systems?"
Exactly, we need to perform CG on larger and more complicated systems, or when we need to watch the dynamics for longer periods of time, and "stimuli-responive systems" would only be one type of system for which we would want to do coarse-graining, and I can imagine some "stimuli-responsive systems" don't require it at all, if you have enough computational resources.
"Unless it is truly rigorous bottom-up CG, I am sometimes skeptical of the forcefields some people use, because the force-field is clearly biased to the problem they are trying to solve. You can coarse-grain frankly any macromolecule, but there has to be a phenomenon some experimentalist is observing that we can't seem to see by plain all-atom MD."
It's fair for you to be skeptical, because some pretty coarse approximations are being made when coarse-graining (hence why it's called "coarse"-graining). I would agree with you that when accuracy is important, it should only be done when necessary (i.e. when doing plain all-atom MD is not worth it). But if accuracy is not important at all, it might even be the case that all-atom MD is like cracking a nut with a sledgehammer.
If you need to know what specific systems/phenomena require simulations that are currently out of reach for all-atom MD and do require coarse-graining, this may be a good question to ask with a title like "what are some interesting examples where coarse-graining was necessary and all-atom MD was infeasible?" with the reference-request
tag (rather than "What kind of systems..."). If you're specifically interested in "stimuli-responsive" systems, you could ask a question with that in the title.