I have no idea if there is anyone today who uses a sheet for sexual intercourse. There is certainly no community that insists on this (not even the Gur Hasidim, whose sexual behavior is extremely ascetic). However, it is simply incorrect to say that this action is not advocated in any text. Usually, when you have a widespread rumor like this, there is some basis for it, even if the original source has been distorted.
What is the origin of the idea of sex through a sheet? The Jerusalem Talmud, Yevamot 1:1, states that R. Yose ben Halafta, who performed the levirate marriage with his sister-in-law, had sex in this fashion: דרך סדין בעל. In case the words דרך סדין were not clear to readers, R. Baruch ben Isaac explains in Sefer ha-Terumah, Hilkhot Halitzah (Jerusalem, 1983), p. 46a, that it means that R. Yose ben Halafta made a hole in the sheet:
נקב עשה בסדין דרך מקום ביאה
This was done so that his personal pleasure be reduced and the focus be on the mitzvah. The Talmud records that he had five children with this woman, and it is not clear if he used the sheet throughout their marriage or only in conceiving the first child (see Korban ha-Edah, ad loc., and also R. Hayyim Kanievsky’s commentary).
If this was all we had, it would not be of great significance. All it would show was that one talmudic sage used the “hole in the sheet” method. There is no implication from the passage that anyone else adopted this approach. Yet based on this text, R. Meir of Rothenburg indeed assumed that when it came to levirate marriage this was the general practice among the pious during tannaitic times.
ואפי’ חסידים הראשונים דור התנאים כשהיו מיבמין היו בועלין דרך סדין כדמשמע בירושלמי (פ”ק דיבמות ה”א) ר’ יוסי בר’ חלפתא הי’ בועל יבמתו דרך סדין
Jacob Emden, whose writings include a good deal about sexual matters, assumes that this type of hasidut is only applicable with levirate marriage, but not with one’s wife.
דווקא ביבמה יש מקום לחסידות כזה לא זולת
Marc B. Shapiro, “The Pew Report and the Orthodox Community (and Other Assorted Comments), part 2”, The Seforim Blog (7 August 2014) [http://seforim.blogspot.com/2014/08/the-pew-report-and-orthodox-community.html]