No. Even a felon can apply for a marriage license. Even someone with a warrent could apply.
One does not have to “pay up” taxes in order to get a marriage lisence.
There are two things you are trying to correlate.
First, is what the OP is hinting at which is the “sovereign citizen” cult (for lack of a better word). These people feel that they can extricate themselves from all legitimately owed taxes and do not get any documentation for their children so they cannot be traced by the state. These children, therefore, cannot participate in normative state activities like driving, getting a job and having a bank account because technically they do not exist. They cannot get a marriage license because they do not have a birth certificate–which–in most states that and $10 will get you a marriage license.
Second, is the fraud that a married couple could perpetrate on the state by living together as two tax-registered individuals rather than a married couple. The state allows tax breaks for those who are single and must maintain a household on their own. For instance, my friend with a child, who has an adult roommate who also has children, can file “head of household” and thus get a discount on what taxes she owes. Her roommate can do the same. Without combining their incomes they are also both eligible for social welfare benefits–although at this time both choose not to take them because having a roommate makes rent cheap enough they can afford groceries and heat.
Now, my husband and I, despite the fact that we look, on the surface, to be demographically similar (two adults + kids) cannot claim this deduction because we are married. If I did not have to combine my income with my husband’s and we each claimed X number of children I would get to be on every social welfare program there is, pay no tax etc. His income isn’t really all that high and while he would not be eligible for any welfare programs he would get more back in tax. In fact, if I claimed to pay rent to him, and charged him for “daycare” while keeping one of the children under my name, he could write off all that daycare expense, I’d not have enough income to pay tax on and we’d make out like bandits. To not be married would TOTALLY game the US system.
This is because the government RIGHTLY assumes that it is more difficult for a single parent than married ones.