Bottom line: This is an urban legend based on some bad (and long outdated) science.
These types of claims come in many forms, using a coin, ice cube, pen, marker, file, and so on, to purportedly cause blistering or second-degree burns on subjects who believe that they are being touched by some implement (hot coal, shell fragment, branding iron, etc).
A common theme of these stories is that they are a recounting of "legitimate" experiments by "legitimate" researchers. Another common theme is the use of suggestion or hypnosis.
Hypnosis is pseudoscience, so this claim is extraneous. Published research is not usually difficult to find, so it ought to be easy enough to confirm such claims, and indeed, a fair amount of published case reports (mostly before 1950), studies, and reviews on such phenomenon exists in the literature.
One such review, by Orne & Hammer (1974), in the 15th edition of Encyclopaedia Britannica summarizes:
There is controversy about the hypnotic induction of blisters. Told
that a cool coin touching him is red hot, a subject shows intense
discomfort; sometimes there may be redness and blistering skin.
Generally not reproducible under controlled laboratory conditions with
normal volunteer subjects, psychologically induced blisters seem most
likely to appear only in those prone to convert emotional disturbances
into symptoms of skin disease.
Fromm & Shor (1972) (reprinted in Fromm & Shor (2006)) provide a more thorough review in their book on Hypnosis:
During the past one hundred years, many researchers suggested to their
hypnotic trance subjects that a blister would form on the skin. With
very few exceptions, no skin changes whatsoever were observed (Barber,
1969b, Ch. 9). About twelve researchers reported that suggestions for
blister formation gave rise to cutaneous alterations; some of these
alterations were labeled "blisters". These reports should be viewed
within a broader context ... delineated in a series of reviews
(Barber, 1961b; Gorton, 1949; Pattie, 1941; Paul, 1963; Sarbin, 1956;
Weitzenhogger, 1953). With very few exceptions, the positive findings were obtained ... prior to the advent of rigorous experimental controls ... A study ... (Borelli, 1953) showed
conclusively that the skin change that was produced by suggestion was
dermographism, not a blister.
A more recent review by Nash & Barnier (2008), in their textbook on Hypnosis, comes to the same conclusion:
The formation of blisters has been reviewed by others (Gorton, 1957;
Paul, 1963; Johnson and Barber, 1976; Barber, 1978; Johnson, 1989;
Gauld, 1990), though there have been very few additional studies
conducted since Pattie's  review. Paul (1963) suggested that
one of the major limitations of a number of prior studies was the
failure to consider the possibility of blister formation resulting
from contact dermatitis. ... In summary, the notion that hypnotic
suggestion can produce blisters does not appear to be reliably
validated by controlled research.
Dermographism and contact dermatitis are allergic reactions to contact with materials. In other words, contact with the items listed above can, in some percentage of the population, cause skin reactions that may superficially resemble blisters.