Hi,
The goverment warned that it would not let the strike continue [indefinitely/forever]
Is there any difference between the two choices? Yes.
indefinitely - the idea is 'without any time limit being set'.
forever - the idea is 'until the end of time'.
Consider these two statements.
My mother-in-law is coming to stay with me indefinitely. ie She hasn't told me the date she plans to leave.
My mother-in-law is staying with me forever. ie She will never leave.
However, informally we sometimes use 'forever' to refer merely to something that seems like a very long time.
eg The movie was boring. It went on forever.
Best wishes, Clive