Depending on the context, "play" is probably correct. The problem is that we need an adverb, "cautiously", to modify the verb (that is, to describe how we play). In colloquial English, you may hear the adjective "cautious" used instead of "cautiously", but we cannot use the noun "caution" to qualify a verb.

If you want to describe someone using caution during an activity, play is not a common verb used. Context matters, however. For example- if I pick up an object that could be dangerous, someone would tell me “use with caution” because I am going to use the object. I think “use with caution” is more natural in English.

I have heard "pay extra caution" before. It means "be extra cautious." I have not heard "Play extra caution"

The expression is pay attention/caution not play attention/caution.

No, you should say 'be extra cautious'