Presuming you don't have a "pallet truck scale" easily available for direct readouts...
You might get an approximation with a bathroom scale placed on a flat/level concrete slab at one end of a plank, with the other end of the plank resting on a board (about same height as the scale) atop same-level concrete as the scale, and the tire near the 'board' end of the plank. This is a "bridge" not a "teeter-totter" lever setup.
Measure the distance from the plank to tire centerline, and from plank to scale, which gives you the leverage factor. Assuming the plank can hold the weight, a 500 pound tire at 1' from the supporting board should press on the scale with about 100 pounds at 5' away.
If you think the tire is around 500-600 pounds, try half-and-half distancing to keep scale readings within 250-300 pounds (i.e. within typical scale capacity, and you might even determine the halfway point by eye).
The above method allows estimating the weight of heavy odd objects lacking available specs. Good luck!
(Probably should place the board first, then read the 'tare' weight of the board alone, then put the tire atop it. Subtract that 'tare weight' from the scale reading, before doubling the "tare-subtracted scale reading" for the half-and-half distancing etc. One can also get tweakier with a short 2x4 across the scale to better define the lever arm distance...which can affect your tare reading also.)
The few full-size truck scales I've encountered read-out weight to the nearest 20 lbs, so you can probably achieve equivalent or better precision with this method.
If you have a big flippable tire stored outdoors, as I did, and it fills up with rainwater -- one solution is to get a very strong drill and circular holesaw (maybe 2" diameter) and core-out a piece in the center of the tread. Be advised the rubber can be unbelievably abrasive and grippy, this stripped all the factory white paint off the hole saw. But it did provide an adequately large hole that did not clog thereafter, as long as the tire was stored with that hole directly downward, atop gravel. Eliminating standing water is important if you have mosquitos in your area.
...hopefully the above is more useful than the glib answer to the thread title "How do you find the weight of big tires?": "I find them to be rather heavy, thank you."
Last edited by Excel; 03-31-2013 at 10:50 AM.
Reason: more info