I have a 5210 from the parts manual things look about the same. I have included a URL to the parts manual. A couple questions, with the tractor OFF, lightly with your finger tap on the glass does the needle move? When you start the tractor does the needle move to zero? The tach has an air core meter, if the needle moves to zero good chance the main chip is ok. Do other the gauges in the cluster work ok, along with lights and warning indicators? When the tractor is running does the hour meter increase the hours and not just come on?
I would start at the magnetic pickup (sender, tachometer Re519144 $27.25). In front of the fuel pump. Disconnect the negative side of the battery. Take it out, make sure it is clean of metal pieces and does not appear to be damaged. Make sure the o-ring is there. It only takes 4 pounds of torque to put it back in, do not want it to touch the gear. Next check the wiring. There is 2 wires coming out the top of the sensor. Using an ohm meter test the ground. You will need to remove the instrument cowl and disconnect the plug/wire at the back of the instrument cluster. On my tractor a green wire (325) goes from the sensor to the cluster, you may need to follow the wires back from the sensor to the cluster. From the parts manual the back of your cluster looks different then mine. With the ohm meter test the wire from the sensor to the back of the cluster.
Would not hurt to make sure the battery terminals are clean and the ground for the front wiring harness.
If you made it this far, and things are not working most likely the problem is in the instrument cluster it is RE62203, $440.00. If you decide to replace the cluster, take a picture of the hours as the new cluster will show zero hours. Also if you do replace the cluster and your instrument glass fogs up for you. You may want to replace that, I included a URL on this.
Let me know how it goes and what you decide to do.
John Deere - Parts Catalog
Instrument Panel Fogs Up : Fog forms inside instrument panel on some 5000 series tractors, making it hard to read gauges