The most well-known story would be the torture of Furutaka Shuntaro before the Ikedaya Incident.

The method of interrogation, carried out by Shinsengumi vice-commander Hijikata Toshizō was alleged to be particularly brutal, although it appears to be largely without verification. With the prisoner unresponsive, Hijikata was said to have suspended the man by his ankles, restraining his wrists, and driven five-inch spikes into the heels of the man's feet. Placing lit candles upon the holes, hot wax dripped deep into his calves. The prisoner eventually claimed that they planned to set fire in Kyoto and capture Matsudaira Katamori, the daimyō of the Aizu clan whose duties included policing Kyoto at the time.

(source: Wikipedia)

I was able to verify that this did happen in Nagakura Shinpachi's diary (浪士文久報国記事):

The Shinsengumi took him into custody to their headquarters, where they examined him and found that his true identity was Furutaka Shungoro (Shuntaro). He was questioned again and again, but he would not confess. When they tortured him, he finally confessed to all of them.

The diary entry itself doesn't say that Hijikata tortured him, but the historian Kimura Sachihiko says it was Hijikata in the commentary.

I also learned from this answer that there is a description of the torture in Shinsengumi Tenmatsuki (新選組顛末記) (a memoir written based on interviews with Nagakura):

They chased down Furutaka and took him back to their headquarters in Mibu. Commander Kondo interrogated Furutaka himself, but Furutaka did not say anything, as he had come to Kyoto with the resolve to die. Even when his back was torn open from all the beating he had taken, he kept his eyes closed, gritted his teeth, and did not open his mouth even when he fainted. Hijikata Toshizo, the second-in-command, was at his wits' end and, after much ingenuity, he first tied Furutaka's hands behind his back and then hung him upside down on a beam.

Then he inserted a five-inch nail into the sole of his foot and lit a candle on it. The wax flowed down and a hot, lead-like substance spread from the soles of Furutaka's feet to his shins. Even the determined Furutaka could not stand this persistent and brutal pain, and after struggling for about half a minute, he finally opened his mouth and revealed his comrades’ secret plan.

[Note: I wasn't able to find this paragraph in my copy of Shinsengumi Tenmatsuki, but it was quoted here from this edition, so I might have an incomplete version]

Since these are primary sources from Nagakura, it seems credible that Hijikata did torture Furutaka.

As for other cases of torture, according to this answer:

The only story of horrendous torture is that of Furutaka.

Rather, it's proven that the torture was done because the Shinsengumi had obtained extraordinary information.

There is no story of torture of spies who infiltrated the Shinsengumi.

In addition, Asahigata Kametaro (旭形亀太郎), a Choshu sumo wrestler who was a secret spy, said in a historical discussion meeting that he was captured and tortured by the Miburoshi before he joined the Shinsengumi, and that he was tied up to a pine tree and doused with water.

Incidentally, Kametaro later joined the Miburoshi.

There are no other stories of torture.

If that's true then Hijikata and the Shinsengumi did torture people, but probably not as much as they’re often depicted.

On a side note, here's what I found out about this Shinsengumi member named Asahigata Kametaro, who lived an interesting life...

A spy? A Shinsengumi member from the Choshu Sumo Wrestler Corps?

Me: