Hi, coffeekat!
Yes, urine can leak out of the site after the tube is removed. You should keep a clean dressing over the site to capture the urine that is draining. If there is a lot of urine drainage, notify the doctor. You can also put a colostomy bag over the site to catch the draining urine if there is a lot. That will save you from having to change the dressing all the time. If protecting the skin from breaking down becomes a problem, you should wash and dry the skin around the leaking site and apply zinc oxide ointment which will act as a barrier against the urine. Another type of barrier would be to place something like a Duo-Derm (that's a brand name for a colloidal dressing) over the skin with just enough of a hole cut into it to allow urine to drain through the old puncture wound, then place absorbant dressings on top of it to catch the draining urine.
Welcome to allnurses!