I can't take credit for this idea - one of the servers at a recent wedding suggested it to me and I thought it was adorable. She noticed that my bride & groom had a burlap aisle runner (they work soooo much better in the sand than other runners) and asked me what they were going to do with the runner after the wedding (not knowing it was rented). She suggested cutting it up and using it as the matting in a picture frame and they could put their wedding photo on top of it.
Since most brides rent a runner instead of buying it, it's probably not all the feasible. But a lot of brides have been requesting burlap on their bouquet stems, or they bring their own burlap table runners, etc. With left over burlap, look at the cool things you could make (or buy!):
Of course, custom monogrammed runners are always popular and this trend has moved into the burlap market. You could always frame the monogrammed piece of your runner after the wedding day. This is so pretty, by PinkTulipOfDaphne:
And since I'm on Etsy already, did you see the blue mason jar I posted yesterday on Facebook? My heart swooned. My wallet was like, "yeah, right!".