Why does the transformer burn up, about 2 minutes after i turn on the power to the furnace? Nothing gets turned on at the thermostat. I just plug it in and push in the blower door switch and it burns up. I know after a lightning strike I heard a "slap" sound and then smelled burning. I know it was the transformer because there was no power light on on the control board. When I replaced the transformer the lights came on, then the transformer burned up. So I replaced the contactor and the transformer, the lights came on again, then the transformer burned up.
Your transformer is undoubtedly burning up because something in the rest of the transformer load circuit also got fried in the lightening strike and is placing a load on the transformer that greatly exceeds its normal operating specifications. Could be your control board, could be the thermostat, anything that is a load on the transformer output.
The idea posted in comments to fuse one leg of the transformer output is a good trouble shooting scheme - fuses are going to be cheaper than transformers. So it seems like you will be replacing a few fuses whilst you try to isolate what component is severely loading the transformer. I would place my bet on a relay coil that has burned the insulation on the coil to a near short or a semiconductor that has suffered on chip meltdown to a short.
As others have mentioned, the transformer is burning up because the load on it is too great. This is likely caused by a damaged component within the control circuit, which is drawing far too much current for the transformer. Rather than burning up more transformers, and/or blowing through a crate of fuses, why not try troubleshooting the circuit with the power off.
Turn the power off to the equipment at the breaker or serviceman switch, and insure it's off before poking around in the equipment.
Start by visually inspecting all the low voltage components (relays, circuit boards, limit switches, etc.), looking for obvious signs of damage (broken, burnt, discolored, etc.). Look for things that are out of place, especially things that might be shorted to the chassis or other grounded items.
If you have a thermal imaging camera (not likely), you could look for hot spots when the equipment is powered up. Using an inline fuse, this method might only cost one or two fuses.
If nothing looks bad, it's time to start poking around with a meter. Grab the schematic, and start isolating and testing each component individually.
There is a short somewhere. Typically it is the thermostat wire grounding on the cabinet but after a lightning strike anything could be causing it. I would ohm out all the components to ground. Or you could install fuses on both sides of the transformer and take one thing out of the circuit at a time, when the fuse doesn’t blow that is your problem. I would start by disconnecting all the thermostat wires from the furnace and see if that is the problem.