I don't know of any empirical studies on this, but I'd be willing to be it would have no effect at all.
The evidentiality distinctions languages make are things like this:
- X, and I have direct evidence that X.
- X, as I can infer from indirect evidence.
- X, as hearsay.
- X, according to an authority.
… and not things like:
- X, with statistical significance p < 0.05.
- X, controlling for all of the potential confound factors mentioned in the previous discourse.
The evidential classes actually expressed are sometimes helpful, but when they are, it's obvious, and people already express them, in whatever language. If I write "According to Chomsky," you don't need me to also mark the verb with a particle saying that it's authority-based. If I write "The data suggest," that not only tells you what kind kind of evidentiality we're dealing with, but also tells you that the data only "suggest" rather than "confirm" the hypothesis I'm about to give.
Presumably what you're looking for is encouraging people to deal with evidentiality when it isn't obviously relevant, so they don't make mistakes and propose things they don't actually have sufficient/appropriate evidence for.
But those are exactly the cases where you'd need the evidential classes specific to science, that don't exist in any natural language. When you're reading a paper, there's almost never a question of whether the scientist observed this effect, or just heard about it in the hallway. You know she observed it, and the question is, e.g., whether she did sufficient trials while doing so.
But what if you created a conlang that did have at least part of such an system?
Well, if you believe in a very strong version of Sapir-Whorf, and you're willing to raise children to speak that conlang as a first language, it's conceivable that they would end up paying more attention to significance tests while writing, and therefore be less likely to make unwarranted jumps. I'm pretty skeptical of strong Sapir-Whorf (and I think there are better ways to train people to be scientists than raising them in a language no other scientist speaks…), but again, this is at least conceivable.
However, while I don't think it would have much effect on the quality of the science being done, I can imagine that it might give us useful insights into philosophy of science, and maybe into linguistics.
For example, where is the line between "direct evidence" vs. "infer from indirect evidence"?
As I understand things, when a scout looks ahead and sees the characteristic swirls of a fast current pushing past slow eddies, she'll say "There are-with-direct-evidence rapids ahead", not "are-inferred". Presumably, this tells us that "rapids" doesn't mean a definition in terms of average water velocity (which she didn't directly observe), but some kind of feature collection that includes a mental image of what rapids look like visually (which she did directly observe).
So, what if that same scout, trained as a physicist, is looking at cloud chamber photos to measure kaon decay asymmetricity. When she sees the distinctive twisty trail that almost always means a kaon was created and decayed, will she say "This run produced-with-direct-evidence kaons" or "This run produced-inferred kaons"?
The answer to that would probably give us a different window into her mental model than, say, asking her whether she's (consciously) a realist about subatomic particles.