I'm guessing the "iron" test was actually a hemoglobin test (though I'm certainly no expert in blood donation screens...).
I suspect your nurse is reporting a folk tale rather than any true difference. Unfortunately, medical professionals are not always an excellent source of scientific knowledge.
I looked for papers that have actually compared measurements in the two hands. Here's one:
Patel, A. J., Wesley, R., Leitman, S. F., & Bryant, B. J. (2013). Capillary versus venous haemoglobin determination in the assessment of healthy blood donors. Vox sanguinis, 104(4), 317–323. https://doi.org/10.1111/vox.12006
Capillary fingerstick samples were assayed by HemoCue in 150 donors. Fingerstick samples from two sites, one on each hand, were obtained from a subset of 50 subjects. Concurrent venous samples were tested using both HemoCue and Cell-Dyn devices.
They're taking finger pricks and comparing them to venous samples; the finger pricks are a convenient real-world method, the venous samples are considered to be a "gold standard" reference.
Their result that is most relevant to this question is:
Substantial variability in repeated fingerstick HemoCue results was seen (mean hemoglobin 13.72 vs. 13.70 g/dL, absolute mean difference between paired samples 0.76 g/dL). Hand dominance was not a factor.
So, there's a lot of variability in the measurement, but no actual difference between the hands.
Just speculating: the procedure in the clinic is to test the non-dominant hand because people prefer to have the discomfort in the non-dominant hand. If the test comes back adequate, that's the end of it: blood donation starts. If the test comes back inadequate, they repeat the test in the other hand.
A fair number of these retests are occurring in people who have a "true" hemoglobin level above the threshold, but whose initial test was below the threshold simply due to the variability in the test. So, we have a classic "regression to the mean" situation, and the second test often exceeds the threshold.
Because of the selection bias in which people get retested in the dominant hand, the nurse develops a theory (or it gets spread around among other professionals) that the dominant hand produces a higher reading.