There is no way than to go to a safe place
I would say "There is no other way than to go to a safe place," or "There is no way other than to go to a safe place." I think this is idiomatic here, as using "There is no" + "than" requires the word "other" either before "than" or "way".
There is no way except to go to a safe place
You left me no way than telling this to John
Same issue as first sentence. "You left me no other way than..." or "You left me no way other than..."
We can't do much except to wait
I would say "We can't do much except wait." I don't have grammar research to support this, but I believe it is related to answering a question with the use of the modal + do. It is similar to "What can we do? [We can] wait." but not "What can we do? [We can] to wait. We are omitting a complete sentence but keeping the grammatical form intact. Introducing the infinitive makes the sentence ungrammatical so we won't say it like that.
We can't do anything other than to fight
Same issue as the sentence before. I would say "We can't do anything other than fight."
We can do nothing but to fight
Same issue. I would say "We can do nothing but fight."