Why does the transformer on my furnace keep burning 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.