B737 First Officer reported erroneous airspeed indications during the initial climb out; resulting in a termination of the climb and a diversion to a suitable airport.

Narrative

We took off out of ZZZ and everything was operating normally and it was a routine flight; it wasn't until we passed through 10;000 ft when I asked for Econ speed from the Captain while I was hand flying and it gave us just over 300 kts. We both agreed and executed it but then I saw the speed bug jump; it went into the over speed barber pole which was stuck at 270 kts on both sides so I kept 250 kts. We decided to keep 250 kts as well climbed and troubleshoot the problem. All our airspeed indications were working fine and the gear and flaps were up and we decided to stop our climb at FL230. We ran the unreliable airspeed non normal and couldn't really fine what matched our scenario other than erroneous maximum speed bars. After completing the non normal it did not fix our issue and the Captain called maintenance control to see if they could help but they had nothing new to add to what we could do. The Captain and I decided since we could not climb any higher or fly faster to get around weather out west we were going to divert to ZZZ1 which was along the way. We also had to do an overweight landing because of all the fuel we had on board and tried our best to burn it off by descending; using speed brakes; and configuring early. Everyone was informed of our situation we [requested priority handling] and we made an uneventful landing into ZZZ1.