Love Shakespeare. Hate Navy Pier.
We regularly attend shows at Chicago Shakespeare Theater, which is located within Navy Pier. I am hoping to find a restaurant close to but not IN Navy Pier. We are especially fond of dive-y, craft beer serving places, but I fear there is a dearth of those in the area. Sort of casual, good food, maybe even small plates, reasonable wine list, open late.
How close? Within 3 blocks, there isn't much; I'd go to Fox & Obel, the gourmet food store, which has a cafe as well as a sit-down restaurant now. Another nearby choice is Sol y Nieve for tapas.
Within 6-8 blocks, there are hundreds of places. Coco Pazzo Cafe is an Italian bistro just that side of Michigan Avenue. But I'd walk the extra few blocks to Sable, in the Hotel Palomar. It's casual, Chef Terhune's food is delicious (don't miss the sweet corn creme brulee) and reasonably priced, many of the dishes are available in small as well as full portions, and they have artisanal cocktails as well as the normal full bar. Another option near Sable that fits what you're looking for is Quartino, which specializes in Italian small plates, is casual, open late, etc.
Since you regularly attend shows there, I recommend trying all these places!