Wondering if a possessive apostrophe is correct after 'minutes' in the sentence: "After a few minutes' use, the hairdryer stopped working."? I know it's correct to have one in the sentence "In a few weeks' time, I'll be on holiday." but not sure if it's the same?
Yes, use an apostrophe in both your hairdryer and holiday examples.
You could also write
After a few minutes of use, the hairdryer stopped working
which is the somewhat more common way to say this.