What is an iPhone app with good support for Twitter lists?
I found "Tweetlist 4 for Twitter" ($5) when searching the app store. Here are some potential work-arounds: 


Notifications seem to clear when I click "Notifications" at the bottom of the Twitter app for iOS 10. Try clicking on each notification in succession without clicking on anything in the the actual app. Works most of the time. 
A foolproof solution is force touching the notification (if your phone supports it) so that you preview the tweet without even opening the app (only works for individual tweets - doesn't work for "recommendations" or "highlights").
TweetDeck can be used on the mobile web (as well as desktop web/app). You can add your lists as columns and collapse the sidebar so you can see better. You can also click the share button and press "Add to Home Screen" so the web page functions more like an app (edit: this doesn't work as you can't get past the login page but there is a paid app called "DESKTOP LOGIN for TweetDeck" for $3 but it has bad reviews because it isn't worth the money given that it's basically the same as using it in the browser).
Another idea is to set up SMS Notifications. Settings -> Mobile -> Text Notifications -> Check 1st Box: "Tweets from people you've enabled for mobile notifications". That way you'll have a running feed to make sure you didn't miss anything, I haven't tried this myself. SMS Notifications Twitter Help Page


PS: Disappearing notifications are frustrating but I've always been optimistic that Twitter will fix it in the future! Jack Dorsey, CEO of Twitter, has declared that they will make efforts to improve Twitter in 2017. Hopefully this is one of the fixes.