Is it profitable to offer free shipping for certain items or orders over a certain amount?

Yes, offering a minimum requirement in exchange for free shipping is profitable, but for it to work, the minimum amount to spend should be more than what a customer would usually pay.

However, if you set the minimum amount too high, customers may not be willing to spend that much money. Here's a way of determining the minimum amount to set:

If it is similar to AOV then it will hurt your business. Generally, increase it by 30% above your AOV

To get the right minimum amount you'll have to continue testing different prices.