To delete routes on the iPhone, go to the MAPS view. Tap the FAVORITE/RECENT ROUTES button at the top of the screen. It looks like a star with a clock next to it. The ROUTES view will appear. Tap either FAVORITE or RECENTS to find the route you want to remove. Use the EDIT button or swipe from right to left across the route you want to delete and then select DELETE.
Alternatively, if you want to delete the current route that is displayed on the Maps page, then from the MAPS page, tap the 'x' to the right end of the Search box. That will clear the route and you can enter a new Route if you desire.