For a variety of reasons.
There are a number of inaccurate assumptions in the question. First, I'd like to point out that such a policy would hurt low-income people by reducing their income and forcing them to move, and would be objectionable to many voters on those grounds. Not all politics is driven purely by self-interest: some voters actually care about the well-being of the general population. Moving on to the other considerations....
First, it's often far easier for wealthy people to change locations than for less wealthy people. They're more likely to have a cushion to absorb the costs of not having a job for some time after moving, more likely to have a job offer before moving, more likely to have money for the down payment for an apartment, and overall better able to absorb the economic risks of moving to another location. For people with particularly low incomes, even the cost of transportation to another state with lower taxes or the cost of moving their furniture may be prohibitively high. This gap in economic freedom of movement could render the proposal in the question simply ineffective. Instead, many of the people targeted would remain where they were. They would experience greater hardship because of the additional regressive taxes, of course, which besides being intrinsically bad might increase the rates of certain crimes as the targeted people attempted to gain income through less-than-legal means.
Also, although poverty is correlated with crime, the correlation is not so simple. Some crimes, such as drug use, have less or no correlation with income. The rate of crime among wealth people is likely understimated due to their having better resources for concealment and less police investigation. And the factors that do increase crime in poorer communities are not intrinsic to poor people so much as related to being poor: for instance, theft is a little more attractive when one is starving or at risk of homelessness; a great deal of violence is due to organized crime, which presents an alternative means of income. Reducing poverty would be a more effective means of dealing with these forms of crime than worsening it and hoping that people left.
Perhaps the most flawed assumption is that trying to get everyone below a certain income level to leave a region would be economically sound.
There would be enforcement costs: people would take money under the table instead of paying high taxes. Which means the government has to audit them. Which costs money.
Further, although welfare is a cost, there are other factors to consider. For instance, the value of labor: everyone's labor, including that of poor people, contributes to the economy. That labor is possibly even worth more than the cost of welfare. Effectively kicking out a large portion of a state's labor force could have a negative effect on the economy, or at least offset much of the short-term gains from decreased spending on social safety nets.
There would be additional costs from displacement: wealthy people and poor people don't do the same jobs, but all those jobs need to be done. The cost of retraining wealthy people for jobs that were previously done by less-wealthy people, and the possible higher salaries they would demand, would be significant.
- Along these lines, welfare can actually be economically beneficial in the long term. Just as increased spending on education provides returns in the future, money spent on some welfare programs produces economic growth. Reducing welfare spending is not necessarily reduced costs, in the long term.
Further, as mentioned in the other answers, nothing would prevent a sort of retaliatory tax war among states or nations, with the end result that poor people across a wide geographic area would be subject to punitively high taxes, increasing suffering, poverty, and crime, as well as welfare spending.
Finally, other answers and comments have mentioned the history of poll taxes in Great Britain, and reasons why they'd be unpopular there. Other countries have had similarly bad experiences. In the United States, the country mentioned in the question, poll taxes were used as a condition to vote, that is to say, to systematically disenfranchise black voters and voters from other racial minority groups. This is because exemptions were granted in a manner that favored white voters, and enforcement focused on African-American voters. This sort of history would make many voters—not just low-income voters but people of color and liberals—suspicious of a poll tax, even if the stated goal had nothing to do with disenfranchisement. Due to the wide reach of US media, this precedent may influence even other nations.
Other countries have also had negative experiences with head taxes: for instance, in Indonesia, some head taxes in the Dutch colonial period applied only to ethnically Chinese people, and their similar successor tax regime exempted groups such as Arabs.