Sorry to tell you this, but yes. The oven cleaner residue is probably what's making your throat burn. But take heart - the residue will totally burn away after a few uses of your oven, and no more burning throat. Meantime, you might want to run the exhaust fan or if it isn't too cold, open a kitchen window a crack. Staying out of the kitchen as much as possible while the oven is on for now should also help.
And yes, you can get oven cleaner that has no fumes. It's a lemon base and you can get it at just about any store.
Of course oven cleaner is toxic. If you want it to work, it has to be. But it is a base, like soda or ammonia. Perhaps you should wipe out the oven with vinegar. That should neutralize any remaining oven cleaner that is left in there. I suspect leftover over-cleaner is what you are smelling.
Oven Cleaner is TOXIC.
Here's a better way: Dust the insides of the oven with baking soda. Spritz heartily with water and vinegar, and leave it overnight. Scrub in the morning. It may take longer than oven cleaner, but it will save your lungs and the environment!
I think you answered your own question. Take a damp rag and wipe out the inside of your oven which will help remove any remaining oven cleaner.
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