I renew several items every day, often interspacing them with newly listed items. The newly listed and newly renewed items always get more views.
That said, I never renew the same item more than once a week, and sometimes I leave items un-renewed for the entire time frame just to leave them in the "long time listed" category. I have noticed that sometimes those items getting close to expiring suddenly pop up with a lot of traffic, so I know they have appeared early in the search results.
I especially find that if I renew in the middle of the night (like when I happen to not be able to sleep LOL) that those items will get views in higher proportion during the following few hours than other items in my shop do, and have resulted in more foreign sales.
It could be that since I have both vintage jewelry as well as handmade jewelry that I'm hitting two categories that have a higher "browse" factor, but that I cannot say. I can tell you that it does help and my sales have increased since I started renewing frequently.
Another factor to consider is your number of followers (which you really cannot control - just takes time) because the more followers you have the more often your new items will get hits as they pop up on the top of their activity feed as "new items from your favorite shops". More hits = more favs = more sales.