On the iPhone and iPad, is there an app that can search a webpage or Twitter every few minutes and pop up notifications?
Use IFTTT.

(IFTTT is a cloud service sometime referred to as “If This Then That”.)

Download the IFTTT app to your phone.  I don’t see a pre-made applet that does exactly what you want, but applets are easy to create from scratch.

For your first applet, the IF trigger will be twitter “new tweet from search” and for the “THEN THAT” pick “notifications” and select “push notification from IFTTT app.

That’s enough to get you started, there are many many pre-made apps and as you see the ability to create your own using data from 100’s of services.  You really just need to log in and explore the services available. 

Note:  you can also log into IFTTT.com from any browser, mobile or desktop.  (The only reason you “need” the iOS app is if you want to receive IFTTT push notifications on your phone.)