When one wants to visit the Ohel, the Lubavitcher Rebbes' grave, one is asked to take off his shoes. Halachically, one does not take one's shoes when visiting many other holy sites, e.g. the Kotel. Is this practice Mutar, or is it better not to visit the Ohel?
R' Moshe Dovber Rivkin wrote:
דאדרבא מקום קברי צדיקי נחשב למקום קדוש, וכמ"ש המהרי"ל "דמקום מנוחת הצדיקים הוא מקום קדוש וטהור והתפלה מתקבלת יותר" הביאו הבה"ט בססי׳ תקפ״א.)ובוודאי מטע"ז נוהגיס הרבה בעת השתטחות על קברי צדיקים לחלוץ הנעלים כשנגשים אל הציון
The burial place of the righteous is (as the Maharil writes) "holy and pure, where prayers are accepted". That's why many, when going to prostrate themselves on the graves of the righteous, take off their shoes when they approach the grave.
The Nitei Gavriel also writes that this was the custom of R' Yisrael of Sadigora (the grandson of the Ruzhiner) and Chernobyl.
 : One of the Chossidim of the Rebbe Rashab (the fifth Lubavitcher Rebbe) and the Previous Lubavitcher Rebbe, and one of the Roshei Yeshiva of Torah VaDaas)
It is muter. There is no reason to suspect this would not be allowed according to halacha anymore than removing another inconsequential article of clothing like a pair of gloves or a hat. It is also not obligatory and is only relevant to those who wish to do so. Furthermore, even for those who wish to do so don't go barefoot when they have non leather shoes available to them. When it isn't very crowded there are crocs available for those who wish to change into non leather shoes.
In regards to those who would attempt to insinuate that this is somehow forbidden or otherwise connected to idol worshiping practices you should take a hard look at yourself and ask, "why do I feel the need to put down other frum Jews for no reason"? There are shivim panim laTorah and the derech of chabad is no less valid than any other derech. In fact just as during the generation of the baal shem tov who brought back many from the brink of despair which was a result of the terrible persecution, religious despair from false messiahs, and isolation from the Jewish community today chabad continues this mission uplifting the souls of Jewish people everywhere from the isolation of galus for those who are tinuk shenishba as well as every segment of Jewry because of the importance of loving one's fellow Jew.
The source and reason for doing so is learned from when Hashem tells Moshe Rabenu to remove his shoes because he is standing on holy ground and from when after Yehoshua has the Jewish people circumcised and speaks to an angel he is told to remove his shoes. However, one is still not required to be barefoot and most visitors wear non-leather shoes instead of going in barefoot. Wearing non-leather shoes is a sign of humility when praying in a holy place.
Chabad.org has a nice article about this, with sources in footnotes.