If you just damaged previously unharmed cans and they don’t show signs of leakage, the cans should be perfectly safe, even if you store them at room temperature.
The US Department of Agriculture Food Safety and Inspection Service writes:
If a can containing food has a small dent, but is otherwise in good shape, the food should be safe to eat. Discard deeply dented cans. A deep dent is one that you can lay your finger into. Deep dents often have sharp points. A sharp dent on either the top or side seam can damage the seam and allow bacteria to enter the can. Discard any can with a deep dent on any seam.
So if you inspected the cans and found them ok, I would recommend you
- Store them at room temperature, not in the fridge, and
- Use them in the near future, not keep them for months or years.
Apart from that, I would probably check the inside of the can when open them, in case the inner lining was damaged and especially acidic foods (think canned tomatoes) caused oxidation or corrosion. I would probably discard them, not because of food safety issues, but because it may cause an unpleasant taste.
For cans with significant damage, the rules for opened cans apply: Don’t store the food in open cans, but empty the content in suitable containers and refrigerate.
Edit, as I had a dented can at hand:
Under normal circumstances, I would probably have left this can at the store. But thanks to the recent wave of panic buying, a family member decided a dented can was better than none.
You can see that the edge of the dent doesn’t affect the seams, so no problem with leakage, and no subsequent food safety issues:
Also, the inner lining is intact, so no risk of the acidic content (tomatoes, in this case) corroding the metal. No problem either:
Still, this was the can I decided to pull first from the pantry, only a week or so after it was bought. I don’t know when the damage happened, though. If I had noticed anything beyond a shape change, I would have discarded the whole can and used another.