Are you trying to move the apps on your home screen? To move apps you hold your finger on one until they all start "dancing." At that point you can tap and hold on any app icon and move it on the screen. When you're done you simply hit the Home button on the bottom of the front of the phone to stop the dancing.
No, in the old days when you open phone app you have your favorite contacts, when you select EDIT you can move then up or down defending on the one you want but in my case there is an option to press and hold a contract to move up or down but nothing happens. You cannot change their position.
Rather than trying to move the icons, go to Safari > Bookmarks > Favorites > Edit > and rearrange the order. Whichever one you move to the top of the list will be the first icon and the bottom one will be the last icon.
Yes,3D Touch is buggy with this. With only 2 favorites the top one easily moves down to become the bottom one but the bottom one won't always move up to become the top one. With several favorites I have one somewhat in the middle that won't move at all.
On second thought it appears to be a "speed" thing. After clicking EDIT-button if you click the move bars( triple bars to the right of contact name ) and move up or down instantly you'll have no trouble. If you delay only slightly, 3D Touch engages and brings up a menu for calling, texting face timing or whatever depending on how contact is set up. Bottom line go FAST!