In invitation is not an invoice. People bring gifts because they want to and they are able to. Not because they got a demand letter from someone telling them to show up with gifts and cash. If I got a birthday invitation like that I would just have to return a nice happy birthday card and my regrets.
What is the purpose of that? If it's for a gift, it is just tacky. People don't have to give you anything besides there are many places that will cash a check for you including the bank of the person who issued the check.
If the purpose is because you are having your party at an amusement park or skating rink and they only accept cash then go ahead.
If "cash only" refers to the form of gifts that you would like, then we'd recommend that you leave it off the birthday invitation. Most people who are invited to a birthday party will bring a present even if they're not explicitly told to. In fact, even if you say "no gifts, please", your guests will likely bring a present anyways.
Instead of "cash only", when your guests call or reply to RSVP and if they ask you for gift ideas, then and only then would you offer a suggestion such as "a gift card to [your favorite store] would always be nice".