Can I put my Aveeno Daily Moisturizer on my sunburned face or will that only make it worse?

I would avoid putting anything with fragrance or chemicals on your face. Aloe vera is perfect or unscented lotion. Be very careful never to let this happen again as you are putting yourself at risk of skin cancer and less importantly, wrinkes when you are older. Wear a high SPF all the time when in the sun.