I'm not familiar with tesla coils, but I have some smps experience.
I'd agree with Tcm, tesla primaries no doubt have loads of leakage inductance as the primary is well clear of the secondary, this being the case there will ne large spikes in the primary, looks like the designer has thought of this adding the transorbs, did you put them close to the fets?, and have you checked them after the little incident?
Also your wiring around the power section needs to be fairly hefty and as short as possible.
Is that a Led/photo cell bottom left for an audio in?, photo cells are not particularly great at audio freq's.