There is a trail (aka, Poudre trail) that follows the entire river through Fort Collins, so there shouldn't be any issues with trespassing. Tubing the river though town with a bunch of friends and beer is a local past-time and if you are doing it on a weekend, good chance you'll find others doing it. The trail is intended for the public, so if you decide to do the town run, you'll see a lot of people on tubes, bikes, horses, swimming, running, etc. Like everyone said, the entire section is very calm and there are is a damn here and there (depending on where you start), but most tubers just flow through it since it's not any type of class. However, super easy to portage with the Poudre trail next to the river.
Below is a link to the map of the Poudre trail. The entire section from Lions Park to CSU Environmental learning center is close to 15 miles long. You can always stop at New Belgium or Odell's as you're floating by.