Depending on the situation punintentional means either an intentional pun as in pun intentional or as in an unintentional pun as in a pun on the word unintentional.
1. "Chop chop I need to get going" said the client in a whimsical way. "That was punintentional" laughed the barber.
2. "Did you hear about the fire at the campsite... it was so intense ( in tents)" said man 1. "Well that was a bad pun" said man 2. "Sorry it was punintentional" said man 1 with an aura of jokiness.
When you slip up, in speech or text, and add or remove some letters from a word that results in a pun. This should be an unintentional slip-up, hence the term.
1. "Off to get some lunch. I'll be back in a bite. Whoa. That was punintentional"
2. "I punintentionally told her I'd meat her at my place."