PARIS—The conflict over a French billionaire's decision to apply for Belgian citizenship escalated on Monday after a newspaper printed a front-page banner headline reading "Get lost, you rich jerk," prompting the industrialist to threaten a lawsuit.

The incident has played into a nationwide debate over tax policy, even though Bernard Arnault, chairman and chief executive of luxury-goods giant LVMH Moët Hennessy Louis Vuitton SA, has said he will still pay French taxes.

Mr....