Native speaker here. This is a bit tricky to answer, since I don't have the chemistry to completely parse this sentence. Domain-specific knowledge might make more general advice incorrect. However, a couple thoughts:
"effect/s": It appears that the study was on the "effects... on structure and dynamics". "Structure and dynamics" are two things, so it seems like "effects" would be plural here.
Let's try a counterfactual: "Alice studied the effect of X on the structure of Y". Since "structure" is only a single thing, it makes sense to use the singular "effect" there.
"anion/s": Since the word "various" is used, that means there is more than one, which makes me lean towards plural. "Anions" seems correct here.
"temperature/s": This is the tricky one. It is correct to say, "Bob studied the effect of temperature on the melting of ice cream". That sentence implies that Bob exposed the ice cream to many different temperatures; I don't see how anyone could read that sentence and conclude that Bob only studied one temperature.
However, if you are using the word 'various', the plural is required. "Bob studied the effect of various temperature on the melting of ice cream" is not correct -- "temperatures" should be used there.
Two thoughts. First, in this study, are anions and temperatures connected or independent? If Berne et al. are studying combinations of anion and temperature, then you could rewrite the sentence as:
Berne et al. studied the effects of various combinations of anions and temperatures on...
As I read this sentence, it means that Berte et al. tested "anion #1 at 50 degrees", "anion #1 at 60 degrees", "anion #2 at 50 degrees", and so on -- i.e. combinations of anion and temperature.
If this is the case, then "various" is modifying "combinations of anions and temperatures", so you need the plural for both 'anion' and 'temperature'. (And 'combination', of course.)
However, if the anion and temperature are not connected, then you could probably use "temperature" (singular) and be OK:
Berne et al. studied the effects of various anions and temperature on...
This sentence to me implies that Berne et al. did some tests with anions (but did not vary the temperature) and some separate tests with various temperatures (but no anions).
This sentence feels a little weird, but it's grammatically OK. The sentence might be clearer if you reversed the position of 'anion' and 'temperature':
Berne et al. studied the effects of temperature and various anions on...
'Temperature' implies a range, even though it's singular; and 'various anions' is unambiguously plural.
Note that this last sentence is, again, only correct if the temperature studies and the anion studies were not connected. If the studies involved combinations of different temperatures and anions, then your sentence needs to reflect that or you may mislead your reader.