Cassandra is correct. The sensor can cause irritation and in some cases even scar tissue.
However, it's been my experience that the issue isn't how long the sensor stays in, but rather, keeping the sensor from moving after the adhesive starts to wear off.
I made the mistake of only using tape to keep my sensor attached after the initial Dexcom adhesive wore off. The tape (flexifix) stuck to my body very well, but it allowed the sensor to move ever so slighly. The best way to explain it is that the sensor was acting like a very tiny knife going slightly in/out. It may have only been a millimeter at most, but it was enough to feel about as painful as a bee sting after a while.
Long story short: I use "Crazy Glue" to keep my sensors attached. Call me crazy, but it works extremely well, is very cost effective, and it's widely available. Whenever I want to finally remove the sensor, I just go for a run/workout to get sweaty and then pull the sensor right off without pain. If you search online, you'll find a number of articles discussing using super glue in medical applications. I would decide for yourself if it's something worth trying.
Side note: I change my Dexcom sensors every 12-18 days by choice largely because they seem to be more accurate later in their lifespan. I'm very lucky to live in the US and have good insurance so it's fortunately not the cost that motivates me to make the sensors last longer than 1 week.