If I review a product (using my photos of the product) or write a tutorial (using screenshots) on my website, do I first need the product manufacturer's permission?
Writing and publishing a review of a product does not usually require permission.
Taking and publishing a photograph of a product in conjunction with a review does not usually require permission either, although there are some edge cases. See for example:
Depending upon whether a screenshot shows something protected by copyright, there may also be copyright issues using screenshots. You should check the licensing terms of the software in question to see whether they mention screenshot usage.
Of course, many product/software vendors will see reviews and tutorials as beneficial, and will not object even if there is an infringement argument.