The sentence with correctly placed commas is:
The competition, which wasn't my idea to begin with, has become too intense.
The other sentences contain incorrect comma usage:
-
The competition, that we are involved in, is way too intense. (It should be "The competition that we are involved in is way too intense" without the commas)
-
The intense competition, was not a good idea. (There should be no comma before "was"; it should be "The intense competition was not a good idea.")
-
The person, who started this competition, wasn't thinking clearly. (It should be "The person who started this competition wasn't thinking clearly," without the commas, because it is a defining relative clause.)