Yeah, it's one of those works of art that aim to make you uneasy. Gotta say, they nailed it. It's a horrible show in more ways than I dare to count.

Here are the relevant episodes where digital minds get tortured.
S2E4 White Christmas: A human mind is copied to a computer that runs home automation systems. The copy is essentially coerced into slavery. When the copy refuses run the automation system perfectly, it is tortured with sensory deprivation over an extended period of time. So, basically slavery, customer satisfaction and quarterly profits are the "justification" here.

S4E1 USS Callister: A guy with some serious issues loves to larp in a digital world with copies of other people. His idea of fun involves abusing the copied minds. It's sort of like mental torture for the copies. He is bullied in real life, and this is his way of getting back to the people he hates. Just plain old revenge transformed into mental abuse, if you can call that a "justification". Hey, I never said the reasons were good.

S4E6 black museum: A museum has a copy of a human mind stuck in a toy. This counts as psychological abuse as the mind is unable to act like a normal human in such a restrictive body. You can also get to electrocute a copy of a murderer's mind, which is just literal torture. Sadistic people pay for it. In other words, money "justifies" torture, again. Actually, money justifies just about anything on this show. On the other hand, they are torturing a convicted murderer so... Oh, but the murderer could be innocent, but we don't really know so... This series just doesn't give you an easy way out.

There are also other episodes that feature copied minds, but they don't involve blatant torture. However, that doesn't make those episodes any less awful, because the writers as sick as they are creative.