This can be a hot-button topic online. Many say "no big deal we do it all the time for years and years". And some say "you're going to burn up in a fiery death".
Truth is, there's not a lot of data showing a preponderance of propane fires on the road. We also think that a big factor is how the propane lines are run in relation to the wheels. If the propane lines go over the wheels or are anywhere near the axles, it's a really good idea to shut off the propane while traveling. A blowout can tear things in the adjacent area up quite a bit! This is just our opinion.
It's definitely a safer move to shut off a fuel source while traveling regardless of line proximity to tires. That said, we traveled with our propane fridge on for about a year (our first year RVing) before we upgraded our batteries with enough Amp Hours to handle running the fridge on our inverter for a whole travel day. We now run on 120V AC via the inverter while we travel.