Is there any reason to keep starred items?

1) As stated in a comment, there is a community centre package that requires Gold star crops. (Parsnips/Melons/Pumpkins/Corn, 5 of each)

2) In Fall, there is a festival where you will be able to show off some quality products and gain points based on their quality, so bringing higher quality items will net your more points.

3) Other than that, higher quality food will yield higher health/energy benefits.

4) I am unsure whether a higher quality items makes a "better" gift than the same item but with a lower quality, but that would be another reason to keep some good quality items.

I mostly sell the quality stuff myself and try and keep some good ones for the Fall festival.

As well as the other benefits mentioned, the effect of any gift is multiplied by a factor of 10% for a silver-quality gift, 25% for a gold one and 50% for an iridium (not sure if this holds for disliked/hated gifts, but I assume it does).

So after giving someone a gift they love on a normal day (ie. not their birthday or anything), you'd normally get 80 friendship points, but if it had a gold star, for example, you'd get a total of (125% x 80), or 100 friendship points - that's 2/5 of a heart.