I am a Canadian citizen. I am in the UK on a short term study visa, with the status conferred by an entry stamp. I want to visit Dublin for a few days and come back to London to continue my studies. Can I do so without needing a new stamp, etc?
Two questions appear here: can the OP enter Ireland, and can the OP re-enter the UK after the Irish excursion.
As to the first, I cannot locate any official UK govt guidance. However, this Oxford University page, titled "Travel During Your Stay," addresses the issue:
If you are using Short-Term Student status to study, there is a difference between having a visa in this capacity and having the status conferred by a stamp on arrival. If you have a Short-Term Student visa in your passport, this should be multi entry so you can use it to re-enter the UK during its period of validity. If you did not need to apply for a visa but were stamped in as a short-term student, this status will lapse when you leave the UK and when you return you will need to qualify again as a short term student, so have with you a recent short-term student letter from your college or course organizer, and evidence of your finances.
Because your UK entry was under a stamp on arrival and does not carry a multiple-entry notation, your Short-Term Student visa will terminate when you leave the UK.
You should, therefore, be fully prepared to justify your re-entry when you reach the UK border. Because your first entry was not stamped "multiple entries," it is possible that entry could be denied.
The second question (entry to Ireland) is easier: a Canadian citizen may enter Ireland for tourism for up to 90 days, without a visa. See the Irish Immigration "Do I Need a Visa" page.
Because there is at least some risk of refusal in attempting a second UK entry without a "multiple entry" already in your passport, the safest choice would be to travel to Ireland after your course of study is completed, perhaps on your way home to Canada.