The Heilige Ruzhiner zt'l (Irin Kadishin, Shemini Atzeres) says that very often, disputes and machlokes break out on the Yom Tov Shemini Atzeres – Simchas Torah. One can check this and he will see that this occurs in all places. The yetzer hara is certainly investing effort to incite machlokes on this day, and that is an indication of the importance of maintaining peace on this day.
This is based on the principle that we can learn from the yetzer hara what is important. If the yetzer hara places effort to get us into a machlokes on this holy Yom Tov, this tells us of the importance of maintaining peace on this Yom Tov.
The Gemara (Shabbos 31a) tells that someone came to Hillel and asked that he convert him while he stands על רגל אחת, on one foot. Hillel told him, "What you hate being done to you, don't do to others." The holy Ruzhiner explains that רגל אחת alludes to Shemini Atzeres, which is called רגל בפני עצמו, its own holiday. Therefore, we understand that the ger tzedek asked Hillel to teach him the secret of Shemini Atzeres. Hillel told him to be cautious with hatred and machlokes. "What you hate being done to you, don't do so to others." That encompasses the secret of Shemini Atzeres – Simchas Torah. It is a Yom Tov of peace.
Therefore, during the hakafos, one has to be extra cautious to honor everyone, and not to chas v'shalom cause distress to anyone. The reward of the hakafos could be lost due to this severe aveirah.
Someone asked the Chofetz Chaim zt'l for the deep meaning and purpose of the hakafos. He replied, "When you are dancing with the talis over your head, be careful not to step on another person's foot. You won't even know whose foot you step on to ask him for forgiveness. This is the סוד ההקפות, the deep secret of the hakafos."
One year, on Simchas Torah, Russian soldiers arrived in Zvhil. Among them were many Jewish soldiers, and they needed food and drink. Rebbe Shlomke of Zvhil said to the chasid Reb Yosel Koz'lik zt'l, "Take this sack and go around to the Jewish homes, and collect food for the Jewish soldiers." Reb Yosel agreed, saying that he would do so immediately after the hakafos.
Rebbe Shlomke replied, "The Torah will rejoice even if you aren't here, but the soldiers are hungry now!" Reb Yosel followed his rebbe's orders, and he immediately went door-to-door to gather food for the soldiers. Rebbe Shlomke's face lit up when Reb Yosel returned with the sack filled with food. Rebbe Shlomke took the sack on his shoulders and danced hakafos around the bimah with it. This expressed his deep love and care for his fellow man.
One time, after the hakafos were finished in the beis medresh of the Maharil Diskin zt'l, the Maharil Diskin requested that they conduct another seven hakafos. Everyone was surprised; nothing like this had ever occurred before. They already did seven hakafos; why should they do another set of seven hakafos?
The Maharil Diskin told his son that there was a talmid chacham in the beis medresh whom they had forgotten to honor with a hakafah. Therefore, the Maharil Diskin repeated the seven hakafos, and during the second set of hakafos, he honored this talmid chacham properly. This was a clever solution. Sometimes, asking forgiveness for not honoring someone only makes matters worse. Without any apologies, he conducted seven more hakafos and thereby was able to honor the talmid chacham.
Imrei Pinchas (Shaar 4:4) tells: One year, before the hakafos of Simchas Torah, Rebbe Pinchas Koritzer zt'l told his student, Rebbe Refael of Bershad zt'l, "Until now, we did hakafos with weapons in our hands. These were the four minim (because on Succos, we circled the bimah with the four minim). The Zohar calls the four minim weapons. But now we're doing hakafos empty-handed. Although we will be carrying the holy sefer Torah, it isn't a mitzvah to hold Sifrei Torah, as it was a mitzvah to hold the four minim. Tell me, Refael. How does it look to perform hakafos empty-handed? How can we do hakafos in this manner? The answer is that a person should become holy like the four minim. His heart should become like the esrog, his spine like the lulav, his eyes like the hadasim, his lips like the aravos, and then he can take the sefer Torah and dance with it."
Rebbe Refael of Barshid cried when he heard this because he felt that he wasn't on this level. Rebbe Pinchas Koritzer calmed him, "Don't cry, Refael. I will carry you on my shoulders."
In other words, in Reb Pinchas Koritzer's merit, his student Reb Refael can dance with the Sefer Torah, and it will be like the four minim are holding the Sefer Torah.