I have a friend who is here on S Pass, he is currently working at an accounting firm that has no career progression, and constantly asks him to work overtime without overtime pay. Is this allowed?

I was a foreigner on an S Pass previously too (now a PR). Working overtime without overtime pay is unfortunately very normal and unless the work contract states that extra pay will be given outside working hours, there's not much you can do.

How long has your friend been in Singapore? Length of stay before applying for PR, current salary, as well as achievements in related field count towards getting the PR approved.

Accounting firms generally require employees to work long hours. That's a fact of life especially for fresh grads. It is a learning period so try to learn as much as he can whilst servicing his clients. However, he should look for another job if he thinks that he is being exploited and singled out.
As for PR, it is pretty tough to get it nowadays. So maybe look elsewhere.
Best of Luck