Snopes seems to have a pretty definite "No" in their article on the matter. They state they could not find the law in Iowa's records, and furthermore:
While this "dumb law" has been shared far and wide on the internet, the majority of these outlets fail to provide any evidence that this law actually exists; when a web site does provide a source link, it is typically directed at the web site Dumblaws.com...several laws (including Iowa's alleged kissing law) are presented without proof. This isn't surprising, as Dumblaws.com writes in its FAQ section that the web site is for "entertainment only" and that the laws appearing on the site have not been properly fact-checked.
They also mention that the earliest reference seems to be a 1998 website listing "law jokes", again without reference.
Although Snopes deals specifically with the five minute claim, the mustache portion of your question also lacks any proper citation, and typically appears alongside the five minute claim, so it also would appear not to be a rule on the books.
Just to get at the validity of your second source, in item 20 of its list (where you got your question from) I invested the claims of it being illegal to kiss on a train in Wisconsin due to Statue 218.071, and a similar claim that in Indiana, you may not kiss a woman when wearing a beard or mustache, per Section 51.(a) (the only two state laws cited). Wisconsin's state website has no record of Statue 218.071 existing, nor can I find any reference the cited Indiana law, so I would not say it is a reliable source to begin, rather just aggregating whatever was found on the internet without fact-checking themselves.