I doubt this has been studies as you ask, i.e. effect on learning. Most of the literature seems to hammer/contrast the effects on learning of multiple-choice vs constructed-response exams, e.g. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3433302/
There have been more limited investigations of 3 vs 4/5 choice in tests. One
meta-analysis from 2005 titled "Three Options Are Optimal for Multiple-Choice Items: A Meta-Analysis of 80 Years of Research"
Multiple‐choice items are a mainstay of achievement testing. The need to adequately cover the content domain to certify achievement proficiency by producing meaningful precise scores requires many high‐quality items. More 3‐option items can be administered than 4‐ or 5‐option items per testing time while improving content coverage, without detrimental effects on psychometric quality of test scores. Researchers have endorsed 3‐option items for over 80 years with empirical evidence—the results of which have been synthesized in an effort to unify this endorsement and encourage its adoption.
This meta-analysis has about 300 citations in Google Scholar, so at least people took notice. I don't know how well received its conclusions were. Wikipedia's page Multiple choice doesn't even mention this paper, but the page is rather sparse in citations anyway.
I did find one newer study 2010 (citing the meta-analysis), which confirmed its conclusions:
Overall, three-option items perform equally as well as four-option items. Since three-option items require less time to develop and administer and additional options provide no psychometric advantage, teachers are encouraged to adopt three-option items as the standard on multiple-choice tests.
And another one from 2009, roughly confirming it:
The low frequency of items with three functioning distractors in the four-option items in this study suggests that teachers have difficulty developing plausible distractors for most MCQs. Test items should consist of as many options as is feasible given the item content and the number of plausible distractors; in most cases this would be three. Item analysis results can be used to identify and remove non-functioning distractors from MCQs that have been used in previous tests.
In these papers, the reason why 3-item MC tests are often just as good has mostly to do with who makes the exam (and the time spend developing it) rather than than who takes it.
On the other hand, there's a 2012 paper on listening tests (evaluated with MCs), which somewhat disagrees:
Mean test scores of the three-option tests were significantly higher than those of four- and five-option tests. While no difference was found in mean item discriminations across the three different test formats, reliability coefficients showed inconsistent patterns depending on the number of options and test versions. One possible interpretation of the low correlations among the scores of three test formats is that items with different numbers of options tap into skills other than listening. The findings suggest that statistically, three options may or may not be optimal depending on the point of view taken – from that of the test score users, or from that of the test stakeholders. Test developers must consider multiple statistical, affective, and contextual factors in determining the optimal number of options.
This was supposedly a high-stakes exam, according to the paper title, namely the CSAT. Of the post-2005 papers, this last one has the most substantial literature and arguments review, so it's worth reading its first 4-5 pages.
